# Trigonometry

## AppendixAAnswers to Selected Exercises and Homework Problems

### 1Triangles and Circles1.1Angles and TrianglesHomework 1.1

#### 7.

$$\theta = 108.8\degree$$

#### 9.

$$\alpha = 29\degree$$

#### 11.

$$\beta = 77\degree$$

#### 13.

$$\alpha = 12\degree$$

#### 15.

$$\theta = 65\degree$$

#### 17.

$$\theta = 12\degree$$

#### 19.

$$\psi = 73\degree$$

#### 21.

$$\phi = 88\degree$$

#### 23.

1. $$\displaystyle \phi = 120\degree$$
2. $$\displaystyle \phi = 160\degree$$
3. $$\displaystyle \phi = \alpha + \beta$$
4. An exterior angle is equal to the sum of the opposite interior angles.

#### 25.

$$\theta = 72\degree, \phi = 54\degree$$

#### 27.

$$\theta = 100\degree, \phi = 30\degree$$

#### 29.

1. $$\displaystyle 180\degree$$
2. $$\displaystyle 90\degree$$
3. a right triangle

#### 31.

1. They are base angles of an isosceles triangle.
2. They are base angles of an isosceles triangle.
3. $$\angle OAB$$ corresponds to $$\theta$$ of Problem 29, and $$\angle OBC$$ corresponds to $$\phi$$ of Problem 29.

#### 33.

$$\alpha = 30\degree, \beta = 60\degree$$

#### 35.

$$x = 47\degree, y = 133\degree$$

#### 37.

$$x = 60\degree, y = 15\degree$$

#### 39.

$$x = 100\degree, y = 16\degree$$

#### 41.

$$x = 90\degree, y = 55\degree$$

#### 43.

$$x = 50\degree, y = 80\degree$$

#### 45.

1. $$\displaystyle \angle 1 = \angle 4, \angle 3 = \angle 5$$
2. $$\displaystyle 180\degree$$
3. In the equation $$\angle 4 + \angle 2 + \angle 5 = 180\degree,$$ substitute $$\angle 1$$ for $$\angle 4\text{,}$$ and substitute $$\angle 3$$ for $$\angle 5$$ to conclude that the sum of the angles in the triangle is $$180 \degree\text{.}$$

#### 47.

$$\angle 1 = 130\degree$$ because vertical angles are equal. $$\angle 2 = 50\degree$$ because it makes a straight angle with a $$130\degree$$ angle. $$\angle 3 = 65\degree$$ because it is a base angle of an isosceles triangle whose vertex angle is $$50\degree\text{.}$$ $$\angle 4 = 65\degree$$ for the same reason. $$\angle 5 = 25\degree$$ because it is complementary to $$\angle 4\text{.}$$

### 1.2Similar TrianglesHomework 1.2

#### 1.

$$\triangle PQT \cong \triangle SRT\text{,}$$ $$x=7\text{,}$$ $$y=3, \alpha=18\degree$$

#### 3.

$$\triangle PRE \cong \triangle URN, z=12\text{,}$$ $$\theta = 10\degree\text{,}$$ $$\phi = 70\degree$$

#### 5.

$$\triangle ABT \cong \triangle ABC,$$ so $$AT=AC$$

#### 7.

Similar. Corresponding sides are proportional.

#### 9.

Similar. Corresponding angles are equal.

#### 11.

$$\angle A = 37\degree, \angle B = 37\degree$$

#### 13.

$$h = 12$$

#### 15.

$$p=35$$

#### 17.

$$g=84$$

#### 19.

$$h=30$$

154 feet

1 mile

17.1 square feet

#### 27.

$$y=\frac{12}{17}x$$

#### 29.

$$h=7.5$$

#### 31.

$$c=15$$

#### 33.

$$s=6$$

#### 35.

$$y=\frac{3}{5}x$$

#### 37.

$$y=5+\frac{3}{4}x$$

#### 39.

1. $$\angle B = 70\degree\text{,}$$ $$\angle CAD = 70\degree\text{,}$$ $$\angle DAB = 20\degree$$
2. $$\triangle DBA$$ and $$\triangle DAC.$$ The hypotenuse is $$BC$$ in $$\triangle ABC\text{,}$$ $$BA$$ in $$\triangle DBA\text{,}$$ and $$AC$$ in $$\triangle DAC\text{.}$$ The short leg is $$AB$$ in $$\triangle ABC\text{,}$$ $$DB$$ in $$\triangle DBA\text{,}$$ and $$DA$$ in $$\triangle DAC\text{.}$$ The longer leg is $$AC$$ in $$\triangle ABC\text{,}$$ $$DA$$ in $$\triangle DBA\text{,}$$ and $$DC$$ in $$\triangle DAC\text{.}$$

### 1.3CirclesHomework 1.3

13 miles

10, 10.00

#### 5.

$$4\sqrt{5} \approx 8.94$$

5

#### 9.

$$2\sqrt {5}$$

5

#### 13.

$$~~24.7$$

#### 15.

1. $$\displaystyle \sqrt{(x+3)^2+(y-4)^2}$$
2. $$\displaystyle \sqrt{(x+3)^2+(y-4)^2}=5$$

#### 17.

The distance between the points $$(x,y)$$ and $$(4,-1)$$ is 3 units.

#### 19.

1. $$6\sqrt{2}~$$cm
2. 8.49 cm

#### 21.

1. $$25\pi~$$sq in
2. 78.54 sq in

#### 23.

1. approximation
2. approximation
3. approximation
4. exact

#### 25.

1.  $$x$$ $$-5$$ $$-4$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$y$$ $$0$$ $$\pm 3$$ $$\pm 4$$ $$\pm \sqrt{21}$$ $$\pm 2\sqrt{6}$$ $$\pm 5$$ $$\pm 2\sqrt{6}$$ $$\pm \sqrt{21}$$ $$\pm 4$$ $$\pm 3$$ $$0$$

#### 27.

1. $$\displaystyle x^2 + y^2 = 36$$

#### 29.

1. $$\displaystyle x^2 + y^2 \lt 9$$

#### 31.

1. No real value of $$y$$ can satisfy $$x^2 +y^2 = 16$$ unless $$-4 \le x \le 4$$
2. The graph has no points where $$x \gt 4$$ and no points where $$x \lt -4$$

#### 33.

$$\sqrt{10}$$

#### 35.

1. $$\displaystyle 12\pi$$

#### 37.

1. $$\displaystyle 4\pi$$

#### 39.

$$(-2\sqrt{5},-4), (2\sqrt{5},-4)$$

#### 41.

$$P(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}), Q(\dfrac{1}{2}, \dfrac{-\sqrt{3}}{2}), R(\dfrac{-3}{4}, \dfrac{\sqrt{7}}{4}), S(\dfrac{-3}{4}, \dfrac{-\sqrt{7}}{4})$$

#### 43.

1. $$\displaystyle 45\degree$$
2. $$5\pi$$ ft
3. $$50\pi$$ sq ft

#### 45.

1. $$\displaystyle \dfrac{2}{5}$$
2. $$40\pi$$ sq ft
3. $$8\pi$$ ft

#### 47.

1. $$\displaystyle \dfrac{1}{10}$$
2. $$\dfrac{\pi}{10}$$ sq km
3. $$\dfrac{\pi}{5}$$ km

#### 49.

1. $$\displaystyle \dfrac{5}{6}$$
2. $$\dfrac{15\pi}{2}$$ sq m
3. $$5\pi$$ m

2070 miles

#### 53.

1. 54,000 miles
2. 2240 mph

#### 55.

1. $$\displaystyle (x-3)^2 + (y+2)^2 = 36$$
2. $$\displaystyle (x-h)^2 + (y-k)^2 = r^2$$

### 1.4Chapter 1 Summary and ReviewChapter 1 Review Problems

#### 5.

$$\alpha = \beta = \gamma = 60\degree$$

#### 7.

$$\phi = \omega = 79\degree$$

#### 9.

$$\theta = 65\degree\text{,}$$ $$\phi = 25\degree$$

#### 11.

$$\delta = 30\degree\text{,}$$ $$\gamma = 60\degree$$

#### 13.

$$\sigma = 39\degree\text{,}$$ $$\omega = 79\degree$$

#### 15.

$$\alpha = 51\frac{3}{7}\degree\text{,}$$ $$\beta = 64\frac{2}{7}\degree$$

#### 17.

$$\triangle ABC \cong \triangle EDC\text{,}$$ $$\alpha = 40\degree\text{,}$$ $$\beta = 130\degree\text{,}$$ $$x=32$$

#### 19.

Yes, three pairs of equal angles

#### 21.

Yes, three pairs of equal angles

13

18

#### 27.

$$y=\dfrac{5x}{2}$$

#### 29.

$$y=\dfrac{7x}{3}$$

#### 31.

$$y=\dfrac{x}{3}$$

#### 33.

$$y=\dfrac{12x}{5}$$

#### 35.

$$\alpha=70\degree$$

14 ft

#### 39.

$$3\frac{3}{4}$$ in

#### 41.

All side have length $$\sqrt{61},$$ opposite sides have slopes $$\dfrac{5}{6}$$ and $$\dfrac{-6}{5}$$

#### 43.

$$AC=BC=18$$

#### 45.

1. $$\displaystyle \sqrt{(x-2)^2+(y-5)^2}=3$$
2. $$\displaystyle (x-2)^2+(y-5)^2=9$$

#### 47.

$$4\sqrt{5} \approx 8.944$$ cm

#### 49.

$$(\dfrac{-1}{3}, \dfrac{2\sqrt{2}}{3}), (\dfrac{-1}{3}, \dfrac{-2\sqrt{2}}{3})$$

#### 51.

1. $$4\pi$$ ft
2. $$\displaystyle 20\pi~ \text{ft}^2$$

#### 53.

1. $$\displaystyle 45\degree, 60\degree$$
2. $$\dfrac{49\pi}{8}~ \text{in}^2, 6\pi~ \text{in}^2$$ Delbert
3. $$\dfrac{79\pi}{4}$$ in, $$2\pi$$ in, Francine

### 2The Trigonometric Ratios2.1Side and Angle RelationshipsHomework 2.1

#### 1.

The sum of the angles is not $$180\degree\text{.}$$

#### 3.

The exterior angle is not equal to the sum of the opposite interior angles.

#### 5.

The sum of the acute angles is not $$90\degree\text{.}$$

#### 7.

The largest side is not opposite the largest angle.

#### 9.

The Pythagorean theorem is not satisfied.

#### 11.

$$5^2 + 12^2 = 13^2\text{,}$$ but the angle opposite the side of length 13 is $$85\degree\text{.}$$

#### 13.

$$4 \lt x \lt 16$$

#### 15.

$$0 \lt x \lt 16$$

21 in

#### 19.

$$6\sqrt{2}~$$in

#### 21.

The rectangle is $$6\sqrt{10}$$ inches by $$18\sqrt{10}$$ inches.

29

#### 25.

$$\sqrt{3}$$

No

Yes

No

#### 33.

The distance from $$(0,0)$$ to $$(3,3)$$ is $$3\sqrt{2}\text{,}$$ and the distance from $$(3,3)$$ to $$(6,0)$$ is also $$3\sqrt{2}\text{,}$$ so the triangle is isosceles. The distance from $$(0,0)$$ to $$(6,0)$$ is 6, and $$(3\sqrt{2})^2 + (3\sqrt{2})^2 = 6^2$$ so the triangle is a right triangle.

25 ft

#### 37.

$$\alpha=30\degree, \beta=60\degree, h=\sqrt{3}$$

#### 39.

$$8\sqrt{3}$$ in

1. No
2. Yes

#### 43.

1. $$(-1,0)$$ and $$(1,0)\text{;}$$ 2
2. $$\sqrt{(p+1)^2+q^2}$$ and $$\sqrt{(p-1)^2+q^2}$$
3. \begin{align*} (\sqrt{(p+1)^2+q^2})^2 \amp + (\sqrt{(p-1)^2+q^2})^2\\ \amp = p^2+2p+1+q^2+p^2-2p+1+q^2\\ \amp =2p^2+2+2q^2=2+2(p^2+q^2)\\ \amp =2+2(1)=4 \end{align*}

### 2.2Right Triangle TrigonometryHomework 2.2

1. 0.91
2. 0.91
3. 0.9063

1. 0.77
2. 0.77
3. 0.7660

#### 5.

1. $$\displaystyle 4\sqrt{13} \approx 14.42$$
2. $$\sin\theta = 0.5547\text{,}$$ $$\cos\theta = 0.8321\text{,}$$ $$\tan\theta = 0.6667$$

#### 7.

1. $$\displaystyle 4\sqrt{15} \approx 15.49$$
2. $$\sin(\theta) = 0.9682\text{,}$$ $$\cos(\theta) = 0.2500\text{,}$$ $$\tan(\theta) = 3.8730$$

#### 9.

1. $$\displaystyle 2\sqrt{67} \approx 16.37$$
2. $$\sin(\theta) = 0.2116\text{,}$$ $$\cos(\theta) = 0.9774\text{,}$$ $$\tan(\theta) = 0.2165$$

14.41

37.86

86.08

#### 27.

1. $$\tan(54.8\degree) = \dfrac{h}{20}\text{,}$$ 170.1 yd

#### 29.

1. $$\tan(36.2\degree) = \dfrac{260}{d}\text{,}$$ 355.2 ft

#### 31.

1. $$\sin(48\degree) = \dfrac{a}{1500}\text{,}$$ 1114.7 m

#### 33.

1. $$\cos(38\degree) = \dfrac{1800}{x}\text{,}$$ 2284.2 m

#### 35.

$$x=\dfrac{82}{\tan(\theta)}$$

#### 37.

$$x=11~\sin(\theta)$$

#### 39.

$$x=\dfrac{9}{cos(\theta})$$

#### 41.

$$36 ~\sin(25\degree) \approx 15.21$$

#### 43.

$$46~ \sin(20\degree) \approx 15.73$$

#### 45.

$$12~ \sin(40\degree) \approx 7.71$$

#### 47.

 $$~~~~$$ sin(\theta) cos(\theta) tan(\theta) $$\theta$$ $$\frac{3}{5}$$ $$\frac{4}{5}$$ $$\frac{3}{4}$$ $$\phi$$ $$\frac{4}{5}$$ $$\frac{3}{5}$$ $$\frac{4}{3}$$

#### 49.

 $$~~~~$$ sin(\theta) cos(\theta) tan(\theta) $$\theta$$ $$\frac{1}{\sqrt{5}}$$ $$\frac{2}{\sqrt{5}}$$ $$\frac{1}{2}$$ $$\phi$$ $$\frac{2}{\sqrt{5}}$$ $$\frac{1}{\sqrt{5}}$$ $$2$$

#### 51.

1. $$\theta$$ and $$\phi$$ are complements.
2. $$\sin(\theta) = \cos(\phi)$$ and $$\cos(\theta) = \sin(\phi)\text{.}$$ The side opposite $$\theta$$ is the side adjacent to $$\phi\text{,}$$ and vice versa.

#### 53.

1. As $$\theta$$ increases, $$\tan(\theta)$$ increases also. The side opposite $$\theta$$ increases in length while the side adjacent to $$\theta$$ remains fixed.
2. As $$\theta$$ increases, $$\cos (\theta)$$ decreases. The side adjacent to $$\theta$$ remains fixed while the hypotenuse increases in length.

#### 55.

As $$\theta$$ decreases toward $$0\degree\text{,}$$ the side opposite $$\theta$$ approaches a length of 0, so sin $$(\theta)$$ approaches 0. But as $$\theta$$ increases toward $$90\degree\text{,}$$ the length of the side opposite $$\theta$$ approaches the length of the hypotenuse, so $$\sin(\theta)$$ approaches 1.

#### 57.

The triangle is not a right tringle.

#### 59.

$$\dfrac{21}{20}$$ is the ratio of hypotenuse to the adjacent side, which is the reciprocal of $$\cos(\theta)\text{.}$$

#### 61.

1. 0.2358
2. sine
3. $$\displaystyle 48\degree$$
4. $$\displaystyle 77\degree$$

#### 63.

1. $$\displaystyle \dfrac{5}{12}$$
2. $$\displaystyle 3$$
3. $$\displaystyle \dfrac{2}{3}$$
4. $$\displaystyle \dfrac{2}{\sqrt{7}}$$

#### 65.

Although the triangles may differ in size, the ratio of the side adjacent to the angle to the hypotenuse of the triangle remains the same because the triangles would all be similar, and hence corresponding sides are proportional.

#### 67.

1. $$\displaystyle \dfrac{2}{3}$$
2. $$\displaystyle \dfrac{2}{3}$$

### 2.3Solving Right TrianglesHomework 2.3

#### 1.

$$A=61\degree, ~a=25.26,~ c=28.88$$

#### 3.

$$A=68\degree, ~a=0.93,~ b=0.37$$

#### 5.

1. $$B=48\degree\text{,}$$ $$~a=17.4\text{,}$$ $$~ b=19.3$$

#### 7.

1. $$A=57\degree\text{,}$$ $$~b=194.4\text{,}$$ $$~ c=357.7$$

#### 9.

1. $$B=78\degree\text{,}$$ $$~b=18.8\text{,}$$ $$~ c=19.2$$

#### 11.

• Solve $$\sin (53.7\degree) = \dfrac{8.2}{c}$$ for $$c\text{.}$$
• Solve $$\tan (53.7\degree) = \dfrac{8.2}{a}$$ for $$a\text{.}$$
• Subtract $$53.7\degree$$ from $$90\degree$$ to find $$A\text{.}$$

#### 13.

• Solve $$\cos (25\degree) = \dfrac{40}{c}$$ for $$c\text{.}$$
• Solve $$\tan (25\degree) = \dfrac{a}{40}$$ for $$a\text{.}$$
• Subtract $$25\degree$$ from $$90\degree$$ to find $$B\text{.}$$

#### 15.

• Solve $$\sin (64.5\degree) = \dfrac{a}{24}$$ for $$a\text{.}$$
• Solve $$\cos (64.5\degree) = \dfrac{b}{24}$$ for $$b\text{.}$$
• Subtract $$64.5\degree$$ from $$90\degree$$ to find $$B\text{.}$$

#### 17.

$$74.2\degree$$

#### 19.

$$56.4\degree$$

#### 21.

$$66.0\degree$$

#### 23.

$$11.5\degree$$

#### 25.

$$56.3\degree$$

#### 27.

$$73.5\degree$$

#### 29.

$$\cos (15\degree) = 0.9659~$$ and $$~\cos^{-1} (0.9659) = 15\degree$$

#### 31.

$$\tan (65\degree) = 2.1445~$$ and $$~\tan^{-1} (2.1445) = 65\degree$$

#### 33.

$$\sin^{-1}(0.6) \approx 36.87\degree$$ is the angle whose sine is $$0.6\text{.}$$ $$(\sin 6\degree)^{-1} \approx 9.5668$$ is the reciprocal of $$\sin (6\degree)\text{.}$$

#### 35.

1. $$\displaystyle \sin (\theta) = \dfrac{1806}{3(2458)},~14.6\degree$$

#### 37.

1. $$\displaystyle \tan \theta = \dfrac{32}{10},~72.6\degree$$

#### 39.

1. $$c = 10\sqrt{10} \approx 31.6\text{,}$$ $$~ A \approx 34.7\degree\text{,}$$ $$~ B \approx 55.3\degree$$

#### 41.

1. $$a = \sqrt{256.28} \approx 16.0\text{,}$$ $$~ A \approx 56.5\degree\text{,}$$ $$~ B \approx 33.5\degree$$

#### 43.

1. $$\tan^{-1}(\dfrac{26}{30}) \approx 40.9\degree,~~91\sqrt{1676} \approx 3612.6$$ cm

#### 45.

1. $$6415$$ km

#### 47.

1. $$462.9$$ ft

(a) and (b)

(a) and (d)

#### 53.

$$\dfrac{\sqrt{3}}{2} \approx 0.8660$$

#### 55.

$$\dfrac{1}{\sqrt{3}} =\dfrac{\sqrt{3}}{3} \approx 0.5774$$

#### 57.

$$1.0000$$

#### 59.

 $$\theta$$ $$~~~0\degree~~~$$ $$~~~30\degree~~~$$ $$~~~45\degree~~~$$ $$~~~60\degree~~~$$ $$~~~90\degree~~~$$ $$\sin (\theta)$$ $$0$$ $$\dfrac{1}{2}$$ $$\dfrac{\sqrt{2}}{2}$$ $$\dfrac{\sqrt{3}}{2}$$ $$1$$ $$\cos (\theta)$$ $$1$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{\sqrt{2}}{2}$$ $$\dfrac{1}{2}$$ $$0$$ $$\tan (\theta)$$ $$0$$ $$\dfrac{1}{\sqrt{3}}$$ $$1$$ $$\sqrt{3}$$ undefined

1. smaller
2. larger
3. larger

#### 63.

$$a = 3\sqrt{3},~b = 3,~B = 30\degree$$

#### 65.

$$a = b = 4\sqrt{2},~B = 45\degree$$

#### 67.

$$e = 4,~f = 4\sqrt{3},~F = 120\degree$$

#### 69.

$$d = 2\sqrt{3},~e = 2\sqrt{2}, f = \sqrt{2} + \sqrt{6}, ~F = 75\degree$$

#### 71.

$$a = 20,~b = 20,~c = 20\sqrt{2}$$

#### 73.

1. $$32\sqrt{3}$$ cm
2. $$128\sqrt{3}$$ sq cm

#### 75.

1. $$10$$ sq cm
2. $$10\sqrt{2}$$ sq cm
3. $$10\sqrt{3}$$ sq cm

#### 77.

1. $$64$$ sq in
2. $$4\sqrt{2}$$ by $$4\sqrt{2}\text{,}$$ area $$32$$ sq in
3. $$\displaystyle 2:1$$

### 2.4Chapter 2 Summary and ReviewChapter 2 Review Problems

#### 1.

If $$C \gt 93\degree\text{,}$$ then $$A+B+C \gt 180\degree$$

#### 3.

If $$A \lt B \lt 58\degree\text{,}$$ then $$A+B+C \lt 180\degree$$

#### 5.

If $$C \gt 50\degree\text{,}$$ then $$A+B+C \gt 180\degree$$

#### 9.

$$a = 97$$

#### 11.

$$c = 52$$

Yes

#### 15.

$$\theta = 35.26\degree$$

#### 17.

No. $$a = 6,~ c = 10$$ or $$a = 9,~ c = 15$$

#### 19.

1. $$\displaystyle w = 86.05$$
2. $$\displaystyle \sin (\theta) = 0.7786,~ \cos(\theta) = 0.6275, ~ \tan (\theta) = 1.2407$$

#### 21.

1. $$\displaystyle y = 16.52$$
2. $$\displaystyle \sin (\theta) = 0.6957,~ \cos (\theta) = 0.7184, ~ \tan (\theta) = 0.9684$$

#### 23.

$$a = 7.89$$

#### 25.

$$x = 3.57$$

#### 27.

$$b = 156.95$$

#### 29.

$$A = 30\degree,~ a = \dfrac{23\sqrt{3}}{3},~ c = \dfrac{46\sqrt{3}}{3}$$

#### 31.

$$F = 105\degree,~ d = 10\sqrt{2},~ e = 20,~ f = 10 + 10\sqrt{3}$$

#### 33.

$$3$$ cm

#### 35.

$$43.30$$ cm

#### 37.

$$15.92$$ m

#### 39.

$$114.02$$ ft, $$37.87\degree$$

#### 41.

1. $$\displaystyle 60.26\degree$$
2. $$\displaystyle 60.26\degree$$
3. $$\displaystyle m = \dfrac{7}{4} = \tan(\theta)$$

#### 43.

1. $$\displaystyle c^2$$
2. $$\displaystyle b - a,~ (b - a)^2$$
3. $$\displaystyle \dfrac{1}{2}ab$$
4. $$\displaystyle 4(\dfrac{1}{2}ab) + (a - b)^2 = 2ab + b^2 - 2ab + a^2 = a^2 + b^2$$

### 3Laws of Sines and Cosines3.1Obtuse AnglesHomework 3.1

#### 1.

1. $$\displaystyle 150\degree$$
2. $$\displaystyle 135\degree$$
3. $$\displaystyle 60\degree$$
4. $$\displaystyle 155\degree$$
5. $$\displaystyle 15\degree$$
6. $$\displaystyle 70\degree$$

#### 3.

1. $$\displaystyle (5,2)$$
2. $$\displaystyle \sqrt{29}$$
3. $$\displaystyle \cos (\theta) = \dfrac{5}{\sqrt{29}},~~\sin (\theta) = \dfrac{2}{\sqrt{29}},~~\tan (\theta) = \dfrac{2}{5}$$

#### 5.

1. $$\displaystyle (-4,7)$$
2. $$\displaystyle \sqrt{65}$$
3. $$\displaystyle \cos (\theta) = \dfrac{-4}{\sqrt{65}},~~\sin (\theta) = \dfrac{7}{\sqrt{65}},~~\tan (\theta) = \dfrac{-7}{4}$$

#### 7.

1. $$\sin (\theta) = \dfrac{9}{\sqrt{97}}\text{,}$$ $$~\cos (\theta) = \dfrac{4}{\sqrt{97}}$$
2. $$\sin (180\degree - \theta) = \dfrac{9}{\sqrt{97}}\text{,}$$ $$~\cos (180\degree - \theta) = \dfrac{-4}{\sqrt{97}}$$
3. $$\displaystyle \theta = 66\degree,~~180\degree - \theta = 114\degree$$

#### 9.

1. $$\sin (\theta) = \dfrac{8}{\sqrt{89}}\text{,}$$ $$~\cos (\theta) = \dfrac{-5}{\sqrt{89}}$$
2. $$\sin (180\degree - \theta) = \dfrac{8}{\sqrt{89}}\text{,}$$ $$~\cos (180\degree - \theta) = \dfrac{5}{\sqrt{89}}$$
3. $$\displaystyle \theta = 122\degree,~~180\degree - \theta = 58\degree$$

#### 11.

1. $$\cos (\theta) = \dfrac{-5}{13}\text{,}$$ $$~\sin (\theta) = \dfrac{12}{13}\text{,}$$ $$~\tan (\theta) = \dfrac{-12}{5}$$
2. $$\displaystyle 112.6\degree$$

#### 13.

1. $$\displaystyle \cos (\theta) = \dfrac{3}{5},~~\tan (\theta) = \dfrac{-3}{4}$$
2. $$\displaystyle 143.1\degree$$

#### 15.

1. $$\sin (\theta) = \dfrac{\sqrt{112}}{11}\text{,}$$ $$~\tan (\theta) = \dfrac{\sqrt{112}}{3}$$
2. $$\displaystyle 74.2\degree$$

#### 17.

1. $$\sin (\theta) = \dfrac{1}{\sqrt{37}}\text{,}$$ $$~\cos (\theta) = \dfrac{-6}{\sqrt{37}}$$
2. $$\displaystyle 170.5\degree$$

#### 19.

1. $$\sin (\theta) = \dfrac{4}{\sqrt{17}}\text{,}$$ $$~\cos (\theta) = \dfrac{1}{\sqrt{17}}$$
2. $$\displaystyle 76.0\degree$$

#### 21.

 $$\theta$$ $$~~~0\degree~~~$$ $$~~~30\degree~~~$$ $$~~~45\degree~~~$$ $$~~~60\degree~~~$$ $$~~~90\degree~~~$$ $$~~~120\degree~~~$$ $$~~~135\degree~~~$$ $$~~~150\degree~~~$$ $$~~~180\degree~~~$$ $$\cos (\theta)$$ $$1$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{1}{2}$$ $$0$$ $$\dfrac{-1}{2}$$ $$\dfrac{1}{\sqrt{2}}~$$ $$\dfrac{-\sqrt{3}}{2}$$ $$-1$$ $$\sin (\theta)$$ $$0$$ $$\dfrac{1}{2}$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{\sqrt{3}}{2}$$ $$1$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{1}{2}$$ $$0$$ $$\tan (\theta)$$ $$0$$ $$\dfrac{1}{\sqrt{3}}$$ $$1$$ $$\sqrt{3}$$ $$\text{undefined}$$ $$-\sqrt{3}$$ $$-1$$ $$\dfrac{-1}{\sqrt{3}}$$ $$0$$

#### 23.

1. $$\displaystyle \sin (\theta) = \sin (180\degree - \theta)$$
2. $$\displaystyle \cos (\theta) = -\cos (180\degree - \theta)$$
3. $$\displaystyle \tan (\theta) = -\tan (180\degree - \theta)$$

#### 25.

1. $$\displaystyle \theta \approx 41.4\degree,~~\phi \approx 138.6\degree$$
2. $$\displaystyle \sin (\theta) = \sin (\phi) = \dfrac{\sqrt{7}}{4}$$

#### 27.

1. $$\displaystyle \theta \approx 81.2\degree,~~\phi \approx 98.8\degree$$
2. $$\displaystyle \sin (\theta) = \sin (\phi) = \dfrac{\sqrt{156279}}{400} \approx 0.9883$$

#### 29.

$$44.4\degree$$ and $$135.6\degree$$

#### 31.

$$57.1\degree$$ and $$122.9\degree$$

#### 33.

$$41.8\degree$$ and $$138.2\degree$$

#### 35.

$$\sin (123\degree) = q\text{,}$$ $$~\cos (33\degree) = q\text{,}$$ $$~\cos (147\degree) = -q$$

#### 37.

$$\cos (106\degree) = -m\text{,}$$ $$~\sin (16\degree) = m\text{,}$$ $$~\sin (164\degree) = m$$

#### 39.

1. $$\displaystyle (4,3),~ (8,6)$$
2. $$\displaystyle y = \tan^{-1}\left(\dfrac{3}{4}\right) \approx 36.87\degree$$
3. $$\displaystyle (-4,3),~ (-8,6);~ 143.13\degree$$

#### 41.

1. $$b=8$$ in, $$h=3\sqrt{3}$$ in
2. $$12\sqrt{3}$$ sq in

#### 43.

1. $$b=6-\dfrac{3\sqrt{2}}{2}$$ mi, $$h=\dfrac{3\sqrt{2}}{2}$$ mi
2. $$\dfrac {18\sqrt{2} - 9}{4}$$ sq mi

#### 45.

1. $$\displaystyle (-1, \sqrt{3})$$
2. $$\displaystyle (-\sqrt{3}, 3)$$

#### 47.

1. $$\displaystyle (-3,3)$$
2. $$\displaystyle (-\sqrt{5}, \sqrt{5})$$

#### 49.

$$20.71$$ sq m

#### 51.

$$55.51$$ sq cm

#### 55.

$$38.04$$ sq units

#### 57.

$$13,851.3$$ sq ft

#### 59.

1. $$\displaystyle (-74.97, 59.00)$$
2. $$\displaystyle BC = 141.97,~~PC = 59.00$$
3. $$\displaystyle 153.74$$

#### 61.

$$\dfrac{\sqrt{5} - 1}{4}$$

#### 63.

Bob found an acute angle. The obtuse angle is the supplement of $$17.46\degree\text{,}$$ or $$162.54\degree\text{.}$$

#### 65.

1. $$\cos (\theta) = \dfrac{x}{3}\text{,}$$ $$~\sin (\theta) = \dfrac{\sqrt{9 - x^2}}{3}\text{,}$$ $$~\tan (\theta) = \dfrac{\sqrt{9 - x^2}}{x}$$

#### 67.

1. $$\cos (\theta) = \dfrac{-\sqrt{4 - y^2}}{2}\text{,}$$ $$~\sin (\theta) = \dfrac{y}{2}\text{,}$$ $$~\tan (\theta) = \dfrac{-y}{\sqrt{4 - y^2}}$$

#### 69.

1. $$\cos (\theta) = \dfrac{-1}{\sqrt{1+ m^2}}\text{,}$$ $$~\sin (\theta) = \dfrac{-m}{\sqrt{1+ m^2}}\text{,}$$ $$~\tan (\theta) = m$$

### 3.2The Law of SinesHomework 3.2

#### 1.

$$x = 7.85$$

#### 3.

$$q = 33.81$$

#### 5.

$$d = 28.37$$

#### 7.

$$\theta = 30.80\degree$$

#### 9.

$$\theta = 126.59\degree$$

#### 11.

$$\beta = 37.14\degree$$

#### 13.

$$a = 4.09\text{,}$$ $$~c = 9.48\text{,}$$ $$~C = 115\degree$$

#### 15.

$$b = 2.98\text{,}$$ $$~A = 36.54\degree\text{,}$$ $$~B = 99.46\degree$$

#### 17.

$$a = 43.55\text{,}$$ $$~b = 54.62\text{,}$$ $$~C = 99\degree$$

a.
b. 808.1 ft

a.
b. 68.2 km

#### 23.

a.
b.1.23 mi $$+$$ 0.99 mi; 0.22 mi

a.
b. 322.6 m

#### 27.

1. $$\displaystyle 1\degree$$
2. $$\displaystyle 66\degree$$
3. 2617.2 ft
4. 1022.6 ft

#### 29.

540,000 AU $$\approx 8.1\times 10^{13}$$ km

#### 31.

750,000 AU $$\approx 1.1\times 10^{14}$$ km

#### 33.

1. $$\displaystyle \dfrac{3}{2}$$
2. No, $$a$$ is too short.
3. 2
4. 1

1. 1,
2. 0,
3. 2,
4. 1,

#### 37.

1. $$\displaystyle C = 25.37\degree,~B = 114.63\degree,~ b = 16.97$$
2. $$C =58.99\degree,~B = 81.01\degree,~ b = 9.22$$ or $$C = 121.01\degree,~B = 18.99\degree,~ b = 3.04$$
3. no solution
4. 5.14

#### 39.

$$A = 40.44\degree,~B = 114.56\degree$$ or $$A = 139.56\degree,~B = 15.44\degree$$

#### 41.

$$C = 37.14\degree,~A = 93.86\degree$$

#### 43.

1299 yd or 277.2 yd

1. 11.79
2. 24.16
3. 24.16

#### 47.

1. $$\displaystyle \dfrac{1}{2} ab \sin (C)$$
2. $$\displaystyle \dfrac{1}{2} ac \sin (B)$$
3. $$\displaystyle \dfrac{1}{2} bc \sin (A)$$

#### 49.

1. $$\displaystyle b = \dfrac{h}{\sin (A)}$$
2. $$\displaystyle h = a \sin (B)$$
3. $$\displaystyle b = \dfrac{a \sin (B)}{\sin (A)}$$
4. ii

### 3.3The Law of CosinesHomework 3.3

#### 1.

1. $$\displaystyle 74 - 70\cos (\theta)$$
2. 12.78
3. 135.22

#### 3.

1. $$\displaystyle \dfrac{a^2 + c^2 - b^2}{2ac}$$
2. $$\displaystyle -0.4$$

#### 5.

1. $$\displaystyle b^2 - 8 \cos (\alpha)b - 65 = 0$$
2. $$\displaystyle 11.17,~ -5.82$$

7.70

13.44

5.12

#### 13.

$$133.43\degree$$

#### 15.

$$40.64\degree$$

#### 17.

$$A = 91.02\degree,~B = 37.49\degree,~C = 51.49\degree$$

#### 19.

$$A = 34.34\degree,~B = 103.49\degree,~C = 42.17\degree$$

6.30 or 2.70

29.76 or 5.91

16.00

#### 27.

Law of Cosines: $$61^2 = 29^2 + 46^2 - 2\cdot 29 \cdot 46 \cos (\phi)$$

#### 29.

Law of Sines: $$\dfrac{a}{\sin (46\degree)} = \dfrac{16}{\sin (25\degree)}$$

#### 31.

First the Law of Cosines: $$x^2 = 47^2 + 29^2 - 2 \cdot 47 \cdot 29 \cos (81\degree)\text{,}$$ then either the Law of Sines: $$\dfrac{\sin (\theta)}{47} = \dfrac{\sin (81\degree)}{x}$$ or the Law of Cosines: $$47^2 = x^2 + 29^2 - 2 \cdot x \cdot 29\cos (\theta)$$

#### 33.

Law of Cosines: $$9^2 = 4^2 + z^2 - 2\cdot 4 \cdot z \cos (28\degree)\text{,}$$ or use the Law of Sines first to find the (acute) angle opposite the side of length 4, then find the angle opposite the side of length $$z$$ by subtracting the sum of the known angles from $$180\degree\text{,}$$ then using the Law of Sines again.

#### 35.

1. $$b = 16.87\text{,}$$ $$~ A = 85.53\degree\text{,}$$ $$~C = 47.47\degree$$

#### 37.

1. $$A = 58.41\degree\text{,}$$ $$B = 48.19\degree\text{,}$$ $$C = 73.40\degree$$

#### 39.

1. $$a = 116.52\text{,}$$ $$~ A = 85.07\degree\text{,}$$ $$~C = 56.93\degree$$ or $$a = 37.93\text{,}$$ $$~ A = 18.93\degree\text{,}$$ $$~C = 123.07\degree$$

#### 41.

1. $$a = 7.76\text{,}$$ $$~ b = 8.97\text{,}$$ $$~C = 39\degree$$

1. 1383.3 m

#### 45.

1. 2123 mi, $$168.43\degree$$ east of north

#### 47.

1. $$7.74\degree$$ west of south, 917.9 mi

1. 92.99 ft

#### 51.

$$147.73~ \text{cm}^2$$

10.53

4.08

#### 57.

1. First figure: $$b - x$$ is the base of the small right triangle. Second: $$-x$$ is the horizontal distance between $$P$$ and the $$x$$-axis, so $$b + (-x)$$ or $$b - x$$ is the base of the large right triangle. Third: $$x = 0\text{,}$$ and $$b$$ is the base of a right triangle.
2. First: $$x$$ and $$y$$ are the legs of a right triangle, $$a$$ is the hypotenuse. Second: $$-x$$ and $$y$$ are the legs of a right triangle with hypotenuse $$a\text{.}$$ Third: $$x = 0$$ and $$y = a$$
3. $$\displaystyle x = a \cos (C)$$

#### 59.

\begin{align*} b^2 + c^2 \amp = (a^2 + c^2 - 2ac \cos (B)) + (a^2 + b^2 - 2bc \cos (C))\\ \amp = 2a^2 + b^2 + c^2 - 2a(c \cos (B) + b \cos (C)) \end{align*}
so $$2a^2 = 2a(c \cos (B) + b \cos (C))\text{,}$$ and dividing both sides by $$2a$$ yields $$a = (c \cos (B) + b \cos (C)$$

#### 61.

For the first equation, start with the Law of Cosines in the form
\begin{equation*} a^2 = b^2 + c^2 - 2bc \cos (A) \end{equation*}
Add $$2ab + 2bc \cos (A) - a^2$$ to both sides of the equation, factor the right side, then divide both sides by $$2bc\text{.}$$
For the second equation, start with the Law of Cosines in the form
\begin{equation*} b^2 + c^2 - 2bc \cos (A) = a^2 \end{equation*}
Add $$2bc - b^2 - c^2$$ to both sides of the equation, factor the right side, then divide both sides by $$2bc\text{.}$$

### 3.4Chapter 3 Summary and ReviewChapter 3 Review Problems

#### 1.

$$\dfrac{1}{2},~\dfrac{\pm\sqrt{3}}{2}$$

#### 3.

1. 49.33
2. $$\displaystyle 114\degree$$

#### 5.

1. $$\cos (\theta) = \dfrac{-2}{\sqrt{13}}\text{,}$$ $$~\sin (\theta) = \dfrac{3}{\sqrt{13}}\text{,}$$ $$~\tan (\theta)= \dfrac{-3}{2}$$
2. $$\displaystyle \theta = 123.7\degree$$

#### 7.

1. $$\cos (\theta) = \dfrac{-4}{5}\text{,}$$ $$~\sin (\theta) = \dfrac{3}{5}\text{,}$$ $$~\tan (\theta) = \dfrac{-3}{4}$$
2. $$\displaystyle \theta = 143.1\degree$$

#### 9.

1. $$\cos (\theta) = \dfrac{-\sqrt{11}}{6}\text{,}$$ $$~\sin (\theta) = \dfrac{5}{6}\text{,}$$ $$~\tan (\theta) = \dfrac{-5}{\sqrt{11}}$$
2. $$\displaystyle \theta = 123.6\degree$$

#### 11.

1. $$\cos (\theta) = \dfrac{-7}{25}\text{,}$$ $$~\sin (\theta) = \dfrac{24}{25}\text{,}$$ $$~\tan (\theta) = \dfrac{-24}{7}$$
2. $$\displaystyle \theta = 106.3\degree$$

#### 13.

$$9.9\degree\text{,}$$ $$~ 170.1\degree$$

#### 15.

$$22.0\degree,~ 158.0\degree$$

#### 17.

1. $$\displaystyle 7\sqrt{2}$$
2. $$\displaystyle 28\sqrt{2}$$

5127.39 sq ft

#### 21.

$$20.41\degree$$

#### 23.

$$a = 27.86$$

#### 25.

$$b = 6.03$$

#### 27.

$$w = 62.10$$

#### 29.

$$s = 15.61~ \text{or}~ 57.45$$

1. 8.82

#### 33.

1. $$\displaystyle 32.57\degree$$

1. 16.29

#### 37.

1. $$\displaystyle 58.65\degree$$

1. 17.40
or
1. 80.93

#### 41.

1. 16.08 mi, 80.4 mph

1. 72.47

1. 353.32
2. 217.52 m

#### 47.

1. 79.64 m
2. $$\displaystyle 35.2\degree$$
3. 46.12 m

#### 49.

$$6.1\degree$$

4.2

22.25 ft

79,332.6 AU

#### 57.

1. $$OW$$ bisects the central angle at $$O\text{,}$$ and the inscribed angle $$\theta$$ is half the central angle at $$O\text{.}$$
2. $$\displaystyle \sin \theta = \dfrac{s}{2r}$$
3. $$\displaystyle r = \dfrac{s}{2 \sin (\theta)}$$
4. $$\displaystyle d = \dfrac{s}{\sin (\theta)}$$

### 4Trigonometric Functions4.1Angles and RotationHomework 4.1

#### 1.

1. $$\displaystyle 216\degree$$
2. $$\displaystyle 108\degree$$
3. $$\displaystyle 480\degree$$
4. $$\displaystyle 960\degree$$

#### 3.

1. $$\displaystyle \dfrac{1}{8}$$
2. $$\displaystyle \dfrac{5}{6}$$
3. $$\displaystyle \dfrac{3}{2}$$
4. $$\displaystyle \dfrac{7}{6}$$

#### 5.

1. $$\displaystyle \dfrac{2}{3}$$
2. $$\displaystyle \dfrac{5}{3}$$

#### 7.

$$60\degree$$

#### 9.

$$60\degree$$

#### 11.

$$14\degree$$

#### 13.

$$400\degree$$ and $$-320\degree$$ (Answers vary.)

#### 15.

$$575\degree$$ and $$-145\degree$$ (Answers vary.)

#### 17.

$$665\degree$$ and $$-55\degree$$ (Answers vary.)

#### 19.

$$295\degree$$

#### 21.

$$70\degree$$

#### 23.

$$315\degree$$

#### 25.

1. $$\displaystyle 36.9\degree,~143.1\degree$$

#### 27.

1. $$\displaystyle 72.5\degree,~287.5\degree$$

#### 29.

$$80\degree$$

#### 31.

$$36\degree$$

#### 33.

$$63\degree$$

#### 35.

$$165\degree\text{,}$$ $$95\degree\text{,}$$ $$345\degree$$

#### 37.

$$140\degree\text{,}$$ $$220\degree\text{,}$$ $$320\degree$$

#### 39.

$$112\degree\text{,}$$ $$248\degree\text{,}$$ $$292\degree$$

#### 41.

$$-0.9205$$

#### 43.

$$-0.7193$$

#### 45.

$$4.705$$

#### 47.

$$-0.7193$$

#### 49.

1. $$\displaystyle 120\degree$$
2. $$\displaystyle 135\degree$$
3. $$\displaystyle 150\degree$$
4. $$\displaystyle 210\degree$$
5. $$\displaystyle 225\degree$$
6. $$\displaystyle 240\degree$$
7. $$\displaystyle 300\degree$$
8. $$\displaystyle 315\degree$$
9. $$\displaystyle 330\degree$$

#### 51.

1. $$\sin (120\degree) = \dfrac{\sqrt{3}}{2},~\cos (120\degree) = \dfrac{-1}{2},~\tan (120\degree) = -\sqrt{3},$$
$$\sin (240\degree) = \dfrac{-\sqrt{3}}{2},~\cos (240\degree) = \dfrac{-1}{2},~\tan (240\degree) = \sqrt{3},$$
$$\sin (300\degree) = \dfrac{-\sqrt{3}}{2},~\cos (300\degree) = \dfrac{1}{2},~\tan (300\degree) = -\sqrt{3}$$

#### 53.

1. $$\sin (135\degree) = \dfrac{1}{\sqrt{2}},~\cos (135\degree) = \dfrac{-1}{\sqrt{2}},~(\tan 135\degree) = -1,$$
$$\sin (225\degree) = \dfrac{-1}{\sqrt{2}},~\cos (225\degree) = \dfrac{-1}{\sqrt{2}},~\tan (225\degree) = 1,$$
$$\sin (315\degree) = \dfrac{-1}{\sqrt{2}},~\cos (315\degree) = \dfrac{1}{\sqrt{2}},~\tan (315\degree) = -1$$

1. III and IV
2. II and III
3. I and III

#### 57.

1. $$\displaystyle 0\degree~ \text{and}~ 180\degree$$
2. $$\displaystyle 90\degree~ \text{and}~ 270\degree$$

#### 59.

$$105\degree$$

#### 61.

$$264\degree$$

#### 63.

$$313\degree$$

#### 65.

Sides of similar triangles are proportional.

### 4.2Graphs of Trigonometric FunctionsHomework 4.2

#### 5.

1. $$\displaystyle \left(-225\degree, \dfrac{1}{\sqrt{2}}\right)$$
2. $$\displaystyle \left(-135\degree, \dfrac{-1}{\sqrt{2}}\right)$$
3. $$\displaystyle (-90\degree, -1)$$
4. $$\displaystyle \left(45\degree, \dfrac{1}{\sqrt{2}}\right)$$
5. $$\displaystyle (180\degree, 0)$$
6. $$\displaystyle \left(315\degree, \dfrac{-1}{\sqrt{2}}\right)$$

#### 7.

1. $$\displaystyle \left(-240\degree, \dfrac{-1}{2}\right)$$
2. $$\displaystyle \left(-210\degree, \dfrac{-\sqrt{3}}{2}\right)$$
3. $$\displaystyle \left(-60\degree, \dfrac{-1}{2}\right)$$
4. $$\displaystyle \left(30\degree, \dfrac{\sqrt{3}}{2}\right)$$
5. $$\displaystyle \left(120\degree, \dfrac{-1}{2}\right)$$
6. $$\displaystyle \left(270\degree, 0\right)$$

#### 9.

1.  $$\theta$$ $$0\degree$$ $$90\degree$$ $$180\degree$$ $$270\degree$$ $$360\degree$$ $$f(\theta)$$ $$0$$ $$1$$ $$0$$ $$-1$$ $$0$$
2.  $$\theta$$ $$0\degree$$ $$90\degree$$ $$180\degree$$ $$270\degree$$ $$360\degree$$ $$f(\theta)$$ $$1$$ $$0$$ $$-1$$ $$0$$ $$1$$

#### 11.

$$\dfrac{7}{2}$$

#### 13.

$$-2\sqrt{2} - 1$$

#### 15.

$$2$$

#### 17.

$$\dfrac{21}{2}$$

#### 23.

1.  $$\theta$$ $$81\degree$$ $$82\degree$$ $$83\degree$$ $$84\degree$$ $$85\degree$$ $$86\degree$$ $$87\degree$$ $$88\degree$$ $$89\degree$$ $$\tan (\theta)$$ $$6.314$$ $$7.115$$ $$8.144$$ $$9.514$$ $$11.43$$ $$14.301$$ $$19.081$$ $$28.636$$ $$57.29$$
2. $$\displaystyle \tan (\theta)~ \text{approaches}~ \infty$$
3.  $$\theta$$ $$99\degree$$ $$98\degree$$ $$97\degree$$ $$96\degree$$ $$95\degree$$ $$94\degree$$ $$93\degree$$ $$92\degree$$ $$91\degree$$ $$\tan (\theta)$$ $$-6.314$$ $$-7.115$$ $$-8.144$$ $$-9.514$$ $$-11.43$$ $$-14.301$$ $$-19.081$$ $$-28.636$$ $$-57.29$$
4. $$\displaystyle \tan (\theta)~ \text{approaches}~ -\infty$$
5. The calculator gives an error message because $$\tan (90\degree)$$ is undefined.

#### 25.

$$y = 6 \sin (\theta)$$

#### 27.

$$y = \cos (\theta) - 5$$

#### 29.

$$y = \sin (4\theta)$$

#### 37.

$$A(0\degree, -3)\text{,}$$ $$~B\left(135\degree, \dfrac{3}{\sqrt{2}}\right)\text{,}$$ $$~C\left(300\degree, \dfrac{-3}{2}\right)$$

#### 39.

$$P(112.5\degree, 1)\text{,}$$ $$~Q(180\degree, 0)\text{,}$$ $$~R(337.5\degree,-1)$$

#### 41.

$$X\left(45\degree, -3 + \dfrac{1}{\sqrt{2}}\right)\text{,}$$ $$~Y(90\degree, -3)\text{,}$$ $$~Z(300\degree,-2)$$

#### 43.

amp$$= 4\text{,}$$ period $$= 360\degree\text{,}$$ midline: $$y = 3$$

#### 45.

amp$$= 5\text{,}$$ period $$= 180\degree\text{,}$$ midline: $$y = 0$$

#### 47.

amp$$= 3\text{,}$$ period $$= 120\degree\text{,}$$ midline: $$y = -4$$

#### 49.

1. amp $$=1\text{,}$$ period $$=90\degree\text{,}$$ midline: $$y = 0$$
2. $$\displaystyle y = \sin (4\theta)$$

#### 51.

1. amp $$=1\text{,}$$ period $$=360\degree\text{,}$$ midline: $$y = 3$$
2. $$\displaystyle y = 3 + \cos (\theta)$$

#### 53.

1. amp $$=4\text{,}$$ period $$=360\degree\text{,}$$ midline: $$y = -2$$
2. $$\displaystyle y = -2 + 4\sin (\theta)$$

#### 55.

1. amp $$=2\text{,}$$ period $$=120\degree\text{,}$$ midline: $$y = 2$$
2. $$\displaystyle y = 2 + 2\cos (3\theta)$$

#### 57.

$$y = -4 + 6 \sin (3\theta)$$ (Answers vary)

#### 59.

$$y = 3 + 2 \cos (\theta)$$ (Answers vary)

#### 61.

$$y = 12 \cos (2\theta)$$ (Answers vary)

#### 63.

$$y = 2 + 5\cos (\theta)$$

#### 65.

$$y = -4\sin (\theta)$$

### 4.3Using Trigonometric FunctionsHomework 4.3

#### 1.

$$36.9\degree,~143.1\degree$$

#### 3.

$$72.5\degree,~287.5\degree$$

#### 5.

$$191.5\degree,~348.5\degree$$

#### 7.

$$154.2\degree,~205.8\degree$$

#### 9.

$$83\degree,~263\degree$$

#### 11.

$$23\degree,~337\degree$$

#### 13.

$$265\degree,~275\degree$$

#### 15.

$$156\degree,~204\degree$$

#### 17.

$$246\degree,~294\degree$$

#### 19.

$$149\degree,~329\degree$$

#### 21.

1. $$\displaystyle (-0.94, -0.34)$$
2. $$\displaystyle (-1.88, -0.68)$$

#### 23.

1. $$\displaystyle (-0.94, 0.34)$$
2. $$\displaystyle (-1.88, 0.68)$$

#### 25.

$$(4\sqrt{2},-4\sqrt{2})$$

#### 27.

$$(-10,-10\sqrt{3})$$

#### 29.

$$(\dfrac{-15\sqrt{3}}{2},\dfrac{15}{2})$$

#### 31.

$$(-1.25,-5.87)$$

#### 33.

$$(5.70, -11.86)$$

#### 35.

$$(9.46,-3.26)$$

#### 37.

1. 15.3 mi east, 21 mi north

#### 39.

1. 91.9 km west, 77.1 km south

#### 41.

1. 30.9 km west, 8.3 km north

#### 43.

$$51.34\degree$$

#### 45.

$$159.44\degree$$

#### 47.

$$y + 5 = (\tan 28\degree)(x - 3)~$$ or $$~y + 5 = 0.532(x - 3)$$

#### 49.

$$y - 12 = (\tan 112\degree)(x + 8)~$$ or $$~y - 12 = -2.475(x + 8)$$

not periodic

#### 53.

Periodic with period 4

1. 10 minutes

1. 1 week

#### 59.

1. period 1 sec, midline $$y = 12\text{,}$$ amp 10 inches

#### 61.

1. period 1 year, midline $$y = 3500\text{,}$$ amp 2500

#### 63.

1. period 1 year, midline $$y = 51\text{,}$$ amp 21

#### 65.

a. IV b. III c. II d. I

#### 69.

1. Emotional high: Oct 5 and Nov 3, low: Oct 19; Physical high: Sep 30 and Oct 23, low: Oct 12 and Nov 4; Intellectual high: Oct 10, low: Oct 26
2. Emotional: 28 days, physical: 23 days, intellectual: 32 days
3. 5152 days

#### 71.

1. periodic, period 8
2. 4, midline: $$y = 3$$
3. $$\displaystyle k = 8$$
4. $$\displaystyle a = 3,~b = 7$$

#### 73.

1. systolic 120 mm Hg, diastolic 80 mm Hg, pulse pressure 40 mm Hg.
2. $$\displaystyle 93\frac{1}{3}$$
3. 72 beats per minute

#### 75.

1. 69 hours.
2. 2.2 to 3.5
3. The larger dip corresponds to when the brighter star is eclipsed, the smaller dip corresponds to when the dimmer star is eclipsed.

### 4.4Chapter 4 Summary and ReviewChapter 4 Review Problems

#### 1.

$$12\degree$$

#### 3.

1. $$\displaystyle 150\degree,~ -210\degree$$
2. $$\displaystyle 240\degree,~ -120\degree$$
3. $$\displaystyle 160\degree,~ -560\degree$$
4. $$\displaystyle 20\degree,~ -340\degree$$

#### 5.

1. $$\displaystyle I,~60\degree;~ 120\degree,~ 240\degree,~ 300\degree$$
2. $$\displaystyle IV,~25\degree;~ 155\degree,~ 205\degree,~ 335\degree$$
3. $$\displaystyle II,~80\degree;~ 80\degree,~ 260\degree,~ 280\degree$$
4. $$\displaystyle III,~70\degree;~ 70\degree,~ 110\degree,~ 290\degree$$

#### 7.

1.  $$\theta$$ $$30\degree$$ $$60\degree$$ $$90\degree$$ $$120\degree$$ $$150\degree$$ $$180\degree$$ $$210\degree$$ $$240\degree$$ $$270\degree$$ $$300\degree$$ $$330\degree$$ $$360\degree$$ $$f(\theta)$$ $$30$$ $$60$$ $$90$$ $$60$$ $$30$$ $$0$$ $$30$$ $$60$$ $$90$$ $$60$$ $$30$$ $$0$$

#### 9.

$$210\degree,~ 330\degree$$

#### 11.

$$120\degree,~ 240\degree$$

#### 13.

$$45\degree,~ 225\degree$$

#### 15.

$$23\degree,~ 337\degree$$

#### 17.

$$72\degree,~ 252\degree$$

#### 19.

$$163\degree,~ 277\degree$$

#### 21.

$$221.81\degree,~ 318.19\degree$$

#### 23.

$$123.69\degree,~ 303.69\degree$$

#### 25.

$$128.68\degree,~ 231.32\degree$$

#### 27.

$$(-9.74, -2.25)$$

#### 29.

$$(-0.28, 8.00)$$

#### 31.

$$(2.84, 0.98)$$

#### 33.

south: 1.74 mi, west: 9.85 mi

#### 35.

$$y = 4 + 7 \sin (180\theta)$$

#### 37.

$$y = 17 + 7 \sin \theta$$

#### 39.

$$\dfrac{\sqrt{3}}{2}$$

0

#### 43.

$$y = 1.5 \cos (\dfrac{\theta}{3}),~ M(-90\degree, \dfrac{3\sqrt{3}}{4}), N(180\degree, \dfrac{3}{4})$$

#### 45.

$$y = 3 + 3 \sin 2\theta,~ A(-45\degree, 6), B(120\degree, 3 - \dfrac{3\sqrt{3}}{2})$$

1. 24 hours

1. 20 sec

#### 51.

1. amp: 2, period: $$360\degree\text{,}$$ midline: $$y = 4$$

#### 53.

1. amp: 3.5, period: $$180\degree\text{,}$$ midline: $$y = 1.5$$

#### 55.

$$30\degree$$

#### 57.

$$92.05\degree$$

#### 59.

$$y = x + 2$$

#### 61.

$$y = -\sqrt{3} x + 3\sqrt{3} - 4$$

#### 63.

The $$\theta$$-intercepts of $$\cos \theta$$ occur at the vertical asymptotes of $$\tan \theta\text{.}$$

### 5Equations and Identities5.1Algebra with Trigonometric RatiosHomework 5.1

#### 1.

$$-2$$

#### 3.

$$\dfrac{1}{\sqrt{2}}$$

#### 5.

$$6$$

#### 7.

$$\dfrac{1}{2}$$

#### 9.

$$4$$

#### 11.

$$2$$

#### 13.

$$1$$

#### 15.

$$0$$

#### 17.

1. $$\displaystyle 0.7660$$
2. $$\displaystyle 0.8164$$
3. $$\displaystyle 0.7660$$

#### 19.

1. $$\displaystyle 0.6691$$
2. $$\displaystyle 1.8271$$
3. $$\displaystyle 0.6691$$

#### 21.

1. $$\displaystyle 1$$
2. $$\displaystyle 1$$
3. $$\displaystyle 1$$

#### 23.

1. $$\displaystyle -2x^2 - x$$
2. $$\displaystyle -2\cos^2 (\theta) - \cos (\theta)$$

#### 25.

1. $$\displaystyle 4SC$$
2. $$\displaystyle 4\sin (\theta) \cos (\theta)$$

#### 27.

1. $$\displaystyle 5C^2S^3$$
2. $$\displaystyle 5\cos^2 (\theta) \sin^3 (\theta)$$

#### 29.

$$-2\cos (t) + 2 \cos (t) \sin (t); ~ 0.6360$$

#### 31.

$$\tan (\theta) - \tan (\phi); ~ -56.91$$

#### 33.

$$2\sin (x) \cos (x) - 2\sin (2x); ~ 0$$

No

No

Yes

No

No

#### 45.

1. $$\displaystyle 2x^2 - x$$
2. $$\displaystyle 2\sin^2 (A) - \sin (A)$$

#### 47.

1. $$\displaystyle ab - 3a^2$$
2. $$\displaystyle \tan (A) \tan (B) - 3 \tan^2 (A)$$

#### 49.

1. $$\displaystyle 2C^2 + C - 1$$
2. $$\displaystyle 2\cos^2 (\phi) + \cos (\phi) - 1$$

#### 51.

1. $$\displaystyle a^2 - b^2$$
2. $$\displaystyle \cos^2 (\theta) -\cos^2 (\phi)$$

#### 53.

1. $$\displaystyle 1 - 2T + T^2$$
2. $$\displaystyle 1 - 2\tan (\theta) + \tan^2 (\theta)$$

#### 55.

1. $$\displaystyle T^4 - 4$$
2. $$\displaystyle \tan^4 (\theta) - 4$$

#### 57.

1. $$\displaystyle 3(3m + 5n)$$
2. $$\displaystyle 3\Big(3\cos(\alpha) + 5\cos(\beta)\Big)$$

#### 59.

1. $$\displaystyle 5r(r - 2q)$$
2. $$\displaystyle 5\tan (C) \Big(\tan (C) - 2 \tan (B)\Big)$$

#### 61.

1. $$\displaystyle (3C+1)(3C-1)$$
2. $$\displaystyle \Big(3\cos (\beta) + 1\Big)\Big(3\cos (\beta) - 1\Big)$$

#### 63.

1. $$\displaystyle 2T^2(3T - 4)$$
2. $$\displaystyle 2\tan^2 (A)\Big(3\tan (A) - 4\Big)$$

#### 65.

1. $$\displaystyle (t - 5)(t + 4)$$
2. $$\displaystyle \Big(\tan (\theta) - 5\Big)\Big(\tan (\theta) + 4\Big)$$

#### 67.

1. $$\displaystyle (3c - 1)(c + 1)$$
2. $$\displaystyle \Big(3\cos (B) - 1\Big)\Big(\cos (B) + 1\Big)$$

#### 69.

1. $$\displaystyle (6S + 1)(S - 1)$$
2. $$\displaystyle \Big(6\sin (\alpha) + 1\Big)\Big(\sin (\alpha) - 1\Big)$$

### 5.2Solving EquationsHomework 5.2

#### 1.

$$70\degree$$

#### 3.

$$40\degree$$

#### 5.

I: $$18\degree;$$ II: $$162\degree;$$ III: $$198\degree;$$ IV: $$342\degree$$

#### 7.

I: $$52\degree;$$ II: $$128\degree;$$ III: $$232\degree;$$ IV: $$308\degree$$

#### 9.

1. $$\displaystyle 0,~4,~2,~0,~4$$
2. $$\displaystyle -1~\text{or}~2$$

#### 11.

1. $$\displaystyle 1,~\dfrac{\sqrt{3}+1}{2},~\sqrt{2},~\dfrac{\sqrt{3}+1}{2}$$
2. $$\displaystyle 45\degree$$

#### 13.

1. $$\displaystyle 0,~\dfrac{2-\sqrt{2}}{2},~\dfrac{1 -\sqrt{3}}{2},~-1$$
2. $$\displaystyle 270\degree$$

#### 15.

$$x = 5,~-3$$

#### 17.

$$x = -3,~1,~2$$

#### 19.

$$\theta = 30\degree ~$$ or $$~ \theta = 210\degree$$

#### 21.

$$\theta = 60\degree ~$$ or $$~ \theta = 300\degree$$

#### 23.

$$\theta = 210\degree ~$$ or $$~ \theta = 330\degree$$

#### 25.

$$\theta = 225\degree ~$$ or $$~ \theta = 315\degree$$

#### 27.

$$\theta = 0\degree ~$$ or $$~ \theta = 180\degree$$

#### 29.

$$\theta = 60\degree, ~\theta = 120\degree,~\theta = 240\degree,~$$ or $$~ \theta = 300\degree$$

#### 31.

$$\theta = 45\degree,~\theta = 135\degree,~\theta = 225\degree, ~$$ or $$~ \theta = 315\degree$$

#### 33.

$$\theta = 104.04\degree ~$$ or $$~ \theta = 284.04\degree$$

#### 35.

$$\theta = 53.13\degree ~$$ or $$~ \theta = 306.87\degree$$

#### 37.

$$\theta = 188.21\degree ~$$ or $$~ \theta = 351.79\degree$$

#### 39.

$$A = 135\degree ~$$ or $$~ A = 315\degree$$

#### 41.

$$\phi = 210\degree ~$$ or $$~ \phi = 330\degree$$

#### 43.

$$B = 90\degree ~\text{or}~ B = 270\degree$$

#### 45.

$$\theta = 210\degree ~$$ or $$~ \theta = 330\degree$$

#### 47.

$$t = 202\degree ~$$ or $$~t = 338\degree$$

#### 49.

$$B = 22\degree ~\text{or}~ B = 202\degree$$

#### 51.

$$\phi = 146\degree ~$$ or $$~ \phi = 214\degree$$

#### 53.

$$\theta = 54.74\degree, ~\theta = 125.26\degree,~\theta = 234.74\degree,~$$ or $$~ \theta = 305.26\degree$$

#### 55.

$$\theta = 0\degree\text{,}$$ $$~\theta = 180\degree\text{,}$$ $$~\theta = 191.54\degree,~$$ or $$~ \theta = 348.46\degree$$

#### 57.

$$\theta = 60\degree\text{,}$$ $$~ \theta = 180\degree\text{,}$$ or $$~ \theta = 300\degree$$

#### 59.

$$\theta = 26.57\degree\text{,}$$ $$~\theta = 161.57\degree\text{,}$$ $$~\theta = 206.57\degree\text{,}$$ or $$~ \theta = 341.57\degree$$

#### 61.

$$\theta = 78.69\degree\text{,}$$ $$~\theta = 108.43\degree\text{,}$$ $$~\theta = 258.69\degree\text{,}$$ or $$~ \theta = 288.43\degree$$

#### 63.

$$\theta = 0\degree$$

#### 65.

$$17.22\degree$$

#### 67.

$$35.66\degree$$

### 5.3Trigonometric IdentitiesHomework 5.3

not an identity

not an identity

identity

not an identity

not an identity

not an identity

identity

identity

#### 17.

$$(1 + \sin (w))(1 - \sin (w)) = 1 - \sin^2 (w) = \cos^2 (w)$$

#### 19.

\begin{equation*} \begin{aligned}[t] \Big(\cos (\theta) - \sin (\theta)\Big)^2 \amp = \cos^2 (\theta) - 2\cos (\theta) \sin (\theta) + \sin^2 (\theta)\\ \amp = \Big(\cos^2 (\theta )+ \sin^2 (\theta)\Big) - 2\sin(\theta) \cos (\theta) = 1 - 2\sin(\theta) \cos (\theta)\\ \end{aligned} \end{equation*}

#### 21.

$$\tan (\theta) \cos (\theta) = \dfrac{\sin (\theta)}{\cos (\theta)}\cdot \cos (\theta) = \sin (\theta)$$

#### 23.

\begin{equation*} \begin{aligned}[t] \cos^4 (x) - \sin^4 (x) \amp = \Big(\cos^2 (x) - \sin^2 (x)\Big)\Big(\cos^2 (x) + \sin^2 (x)\Big)\\ \amp = \Big(\cos^2 (x) - \sin^2 (x)\Big)(1) = \cos^2 (x) - \sin^2 (x)\\ \end{aligned} \end{equation*}

#### 25.

$$\dfrac{\sin (u)}{1 + \cos (u)} \cdot \dfrac{1 - \cos (u)}{1 - \cos (u)} = \dfrac{\sin (u)\Big(1 - \cos (u)\Big)}{1 - \cos^2 (u)} = \dfrac{\sin (u)\Big(1 - \cos (u)\Big)}{\sin^2 (u)} = \dfrac{1 - \cos (u)}{\sin (u)}$$

#### 27.

$$1$$

#### 29.

$$1$$

#### 31.

$$\sin^2 (A)$$

#### 33.

$$\tan^2 (z)$$

#### 35.

$$3$$

#### 37.

$$1$$

#### 39.

$$6$$

#### 41.

$$\cos (2\theta)$$

#### 43.

$$\cos (\theta)$$

#### 45.

$$\sin (2t)$$

#### 47.

$$1 + 2\sin (\theta) + \sin^2 (\theta)$$

#### 49.

$$3\cos^2 (\phi) - 2$$

#### 51.

$$\theta = 90\degree, ~\theta = 180\degree, ~\theta = 270\degree$$

#### 53.

$$\theta = 90\degree, ~\theta = 210\degree, ~\theta = 330\degree$$

#### 55.

$$\theta = 210\degree, ~\theta = 330\degree$$

#### 57.

$$\theta = 18.43\degree,~ \theta = 198.43\degree$$

#### 59.

$$\sin (A) = \dfrac{-5}{13},~ \tan (A) = \dfrac{-5}{12}$$

#### 61.

$$\cos (\phi) = \dfrac{-4\sqrt{3}}{7},~ \tan (\phi) = \dfrac{-1}{4\sqrt{3}}$$

#### 63.

$$\sin (\theta) =\dfrac{-1}{\sqrt{5}}\text{,}$$ $$~ \cos (\theta) = \dfrac{2}{\sqrt{5}}$$

#### 65.

$$\sin (\theta) =\dfrac{-3}{5}\text{,}$$ $$~ \cos (\theta) = \dfrac{-4}{5}$$

#### 67.

$$\sin (\theta) =\dfrac{\sqrt{3}}{2}\text{,}$$ $$~ \cos (\theta) = \dfrac{-1}{2}\text{,}$$ $$~ \tan (\theta) = \sqrt{3}$$

#### 69.

$$\sin (\beta) =\dfrac{2}{\sqrt{5}}\text{,}$$ $$~ \cos (\beta) = \dfrac{-1}{\sqrt{5}}\text{,}$$ $$~ \tan (\beta) = -2$$

#### 71.

\begin{equation*} \begin{aligned}[t] \amp \sin (C) =\dfrac{1}{\sqrt{5}},~ \cos (C) = \dfrac{2}{\sqrt{5}},~ \tan (C) = \dfrac{1}{2}\\ \text{or}~~\amp \sin (C) =\dfrac{1}{\sqrt{5}},~ \cos (C) = \dfrac{-2}{\sqrt{5}},~ \tan (C) = \dfrac{-1}{2}\\ \end{aligned} \end{equation*}

#### 73.

$$\dfrac{\tan (\alpha)}{1 + \tan (\alpha)} = \dfrac{\dfrac{\sin (\alpha)}{\cos (\alpha)}}{1 + \dfrac{\sin (\alpha)}{\cos (\alpha)}} \cdot \dfrac{\cos (\alpha)}{\cos (\alpha)} = \dfrac{\sin (\alpha)}{\sin (\alpha) + \cos (\alpha)}$$

#### 75.

$$\dfrac{1 + \tan^2 (\beta)}{1 - \tan^2 (\beta)} = \dfrac{\dfrac{1}{\cos^2 (\beta)}}{1 - \dfrac{\sin^2 (\beta)}{\cos^2 (\beta)}} \cdot \dfrac{\cos^2 (\beta)}{\cos^2 (\beta)} = \dfrac{1}{\cos^2 (\beta) - \sin^2 (\beta)}$$

#### 77.

1. By the distance formula, $$\sqrt{x^2 + y^2} = r\text{,}$$ or $$x^2 + y^2 = r^2\text{.}$$
2. $$\displaystyle \dfrac{x^2}{r^2} + \dfrac{y^2}{r^2} = 1$$
3. $$\displaystyle \left(\dfrac{x}{r}\right)^2 + \left(\dfrac{y}{r}\right)^2 = 1$$
4. $$\displaystyle \Big(\cos (\theta)\Big)^2 + \Big(\sin (\theta)\Big)^2 = 1$$

### 5.4Chapter 5 Summary and ReviewChapter 5 Review Problems

#### 1.

$$\dfrac{-3}{4\sqrt{2}}$$

#### 3.

$$\dfrac{1}{\sqrt{6}}$$

#### 5.

1. $$\displaystyle 0.8660$$
2. $$0.9848;$$ No

#### 7.

1. $$\displaystyle 1.4821$$
2. $$1.4821;$$ Yes

#### 9.

$$5\sin (x) - 2\sin (x) \cos (y) - \cos (y)$$

#### 11.

$$2\tan (\theta) - 10\tan^2 (\theta)$$

Not equivalent

Equivalent

#### 17.

$$2\cos^2 \alpha + \cos \alpha - 6$$

#### 19.

$$\tan^2 (\phi) - 2\tan (\phi) \cos (\phi) + \cos^2 (\phi)$$

#### 21.

$$6\Big(2\sin (3x) - \sin (2x)\Big)$$

#### 23.

$$\Big(1 + 3\tan (\theta)\Big)\Big(1 - 3\tan (\theta)\Big)$$

#### 25.

$$\cos (\alpha) + \sin (\alpha)$$

#### 27.

$$\dfrac{3}{2}$$

#### 29.

$$\dfrac{3\tan (C) + 2}{\tan (C) - 2}$$

#### 31.

$$51.32\degree,~ 308.68\degree$$

#### 33.

$$90\degree\text{,}$$ $$~ 270\degree\text{,}$$ $$~ 120\degree\text{,}$$ $$~ 240\degree$$

#### 35.

$$90\degree\text{,}$$ $$~ 210\degree\text{,}$$ $$~ 330\degree$$

#### 37.

$$30\degree\text{,}$$ $$~ 150\degree\text{,}$$ $$~ 210\degree\text{,}$$ $$~ 330\degree$$

#### 39.

$$0\degree\text{,}$$ $$~ 120\degree\text{,}$$ $$~ 240\degree$$

#### 41.

$$57.99\degree,~ 237.99\degree$$

#### 43.

$$90\degree,~ 270\degree$$

#### 45.

$$33.17\degree$$

Identity

Not an identity

Not an identity

Identity

#### 55.

$$\dfrac{1 - \cos^2 (\alpha)}{\tan (\alpha)} = \sin^2 (\alpha) \cdot \dfrac{\cos (\alpha)}{\sin (\alpha)} = \sin (\alpha) \cos (\alpha)$$

#### 57.

\begin{equation*} \begin{aligned}[t] \dfrac{\dfrac{\sin (\theta)}{\cos (\theta)} - \sin (\theta) \cos (\theta)}{\sin (\theta) \cdot \dfrac{\sin (\theta)}{\cos (\theta)}} \amp = \dfrac{\sin (\theta) - \sin (\theta) \cos^2 (\theta)}{\sin^2 (\theta)}\\ \amp = \dfrac{\sin (\theta) \Big(1 - \cos^2 (\theta)\Big)}{\sin^2 (\theta)} = \dfrac{\sin (\theta) \sin^2 (\theta)}{\sin^2 (\theta)} = \sin (\theta)\\ \end{aligned} \end{equation*}

#### 59.

$$\dfrac{1}{\sin (\theta) \cos (\theta)}$$

#### 61.

$$1$$

#### 63.

$$0$$

#### 65.

$$1$$

#### 67.

$$\dfrac{1}{\cos^2 (\beta)}$$

#### 69.

$$\sin (x)$$

#### 71.

$$\sin (\beta) = \dfrac{-6}{\sqrt{85}},~ \cos (\beta) = \dfrac{-7}{\sqrt{85}},~ \tan (\beta) = \dfrac{6}{7}$$

#### 73.

$$\sin (\alpha) = \dfrac{\sqrt{21}}{5},~ \cos (\alpha) = \dfrac{-2}{5},~ \tan (\alpha) = \dfrac{-\sqrt{21}}{2}$$

#### 75.

$$0\degree,~ 180\degree,~ 270\degree$$

#### 77.

$$135\degree,~ 315\degree$$

#### 79.

$$0\degree,~ 60\degree,~ 180\degree,~ 300\degree$$

#### 81.

$$0\degree,~ 180\degree$$

#### 1.

 Radians $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2 \pi$$ Degrees $$0\degree$$ $$45\degree$$ $$90\degree$$ $$135\degree$$ $$180\degree$$ $$225\degree$$ $$270\degree$$ $$315\degree$$ $$360\degree$$

#### 3.

1. $$\displaystyle 120\degree = \dfrac{2\pi}{3} \text{radians}$$
2. $$\displaystyle 240\degree = \dfrac{4\pi}{3} \text{radians}$$
3. $$\displaystyle 480\degree = \dfrac{8\pi}{3} \text{radians}$$
4. $$\displaystyle 600\degree = \dfrac{10\pi}{3} \text{radians}$$

#### 5.

1. $$\displaystyle 45\degree = \dfrac{\pi}{4} \text{radians}$$
2. $$\displaystyle 135\degree = \dfrac{3\pi}{4} \text{radians}$$
3. $$\displaystyle 225\degree = \dfrac{5\pi}{4} \text{radians}$$
4. $$\displaystyle 315\degree = \dfrac{7\pi}{4} \text{radians}$$

#### 9.

1. $$\displaystyle 0.52$$
2. $$\displaystyle 2.62$$
3. $$\displaystyle 3.67$$
4. $$\displaystyle 5.76$$

#### 13.

$$2.09$$

#### 15.

$$2.62$$

#### 17.

$$0.52$$

#### 19.

$$2.36$$

1. II
2. IV
3. IV
4. I

1. III
2. II
3. I
4. IV

#### 25.

 Radians $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{3}$$ Degrees $$30\degree$$ $$45\degree$$ $$60\degree$$

#### 27.

 Radians $$\dfrac{7\pi}{6}$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{4\pi}{3}$$ Degrees $$210\degree$$ $$225\degree$$ $$240\degree$$

#### 29.

1. $$\displaystyle 1.31$$
2. $$\displaystyle 4.12$$
3. $$\displaystyle 5.71$$

#### 31.

1. $$\displaystyle 45.8\degree$$
2. $$\displaystyle 200.5\degree$$
3. $$\displaystyle 292.2\degree$$

#### 33.

$$5.86~\text{in}$$

#### 35.

$$4.13~\text{m}$$

#### 37.

$$160.42\degree$$

#### 39.

1. $$\displaystyle \dfrac{5\pi}{6}$$
2. $$\displaystyle 32.72~\text{ft}$$

#### 41.

$$\dfrac{8}{67}~\text{radians}~\approx6.84\degree$$

#### 43.

1. $$\displaystyle 33,000\pi\approx 103,672.6~\text{in}$$
2. $$\displaystyle 33,000\pi\approx 103.672.6~\text{in per min}$$

#### 45.

$$170\pi\approx 534.1~\text{m per min}$$

#### 47.

$$(0.2,0.98)\text{,}$$ $$~(0.2,-0.98)$$

#### 49.

$$(0.94,-0.35)\text{,}$$ $$~(-0.94,-0.35)$$

#### 51.

$$\left(\dfrac{-\sqrt{3}}{2}, \dfrac{1}{2}\right)\text{,}$$ $$~\left(\dfrac{-\sqrt{3}}{2}, \dfrac{-1}{2}\right)$$

#### 53.

1.  $$\theta$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$s$$ $$4$$ $$8$$ $$12$$ $$16$$ $$20$$ $$24$$
2. $$\displaystyle m = 4$$
3. Arclength doubles; arclength triples

#### 55.

1. $$\displaystyle \dfrac{\pi}{10}~\text{radians per min}$$
2. $$\displaystyle \dfrac{10\pi}{9}~\text{radians per sec}$$

#### 57.

1. $$\displaystyle \dfrac{\theta}{2\pi}$$
2. $$\displaystyle \dfrac{3}{8},~\dfrac{5}{6},~\dfrac{7}{12}$$

#### 59.

$$32.5~\text{cm}^2$$

### 6.2The Circular FunctionsHomework 6.2

#### 1.

 $$\hphantom{0000}$$ a b c d $$t$$ $$\dfrac{\pi}{4}$$ $$\dfrac{3\pi}{4}$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{7\pi}{4}$$ $$x$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{-1}{\sqrt{2}}$$ $$\dfrac{-1}{\sqrt{2}}$$ $$\dfrac{1}{\sqrt{2}}$$ $$y$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{-1}{\sqrt{2}}$$ $$\dfrac{-1}{\sqrt{2}}$$

#### 3.

 $$\hphantom{0000}$$ a b c d $$t$$ $$\dfrac{\pi}{3}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{4\pi}{3}$$ $$\dfrac{5\pi}{3}$$ $$x$$ $$\dfrac{1}{2}$$ $$\dfrac{-1}{2}$$ $$\dfrac{-1}{2}$$ $$\dfrac{1}{2}$$ $$y$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{-\sqrt{3}}{2}$$ $$\dfrac{-\sqrt{3}}{2}$$

#### 5.

1. $$\displaystyle \sin (0.4) \approx 0.39,~ \cos (0.4) \approx 0.92,~ \tan (0.4) \approx 0.42$$
2. $$\displaystyle \sin (1.2) \approx 0.93,~ \cos (1.2) \approx 0.36,~ \tan (1.2) \approx 2.6$$
3. $$\displaystyle \sin (2) \approx 0.91,~ \cos (2) \approx -0.42,~ \tan (2) \approx -2.2$$

#### 7.

1. $$\displaystyle \sin (2.8) \approx 0.33,~ \cos (2.8) \approx -0.94,~ \tan (2.8) \approx -0.36$$
2. $$\displaystyle \sin (3.5) \approx -0.35,~ \cos (3.5) \approx -0.94,~ \tan (3.5) \approx 0.37$$
3. $$\displaystyle \sin (5) \approx -0.96,~ \cos (5) \approx 0.28,~ \tan (5) \approx -3.3$$

#### 9.

$$t \approx 1.27$$ or $$t \approx 5$$

#### 11.

$$t \approx 3.92$$ or $$t \approx 5.5$$

#### 13.

$$t \approx 2.72$$ or $$t \approx 5.87$$

II

II

III

Negative

Positive

Positive

#### 27.

$$\sin (3.5)\text{,}$$ $$\sin (0.5)\text{,}$$ $$\sin (2.5)\text{,}$$ $$\sin (1.5)$$

#### 29.

$$\cos (3)\text{,}$$$$\cos (4)\text{,}$$ $$\cos (2)\text{,}$$ $$\cos (5)$$

#### 31.

January 1: 4:24, April 1: 6:45, July 1: 8:02, October 1: 5:55

#### 33.

$$1.34$$

#### 35.

$$0.84$$

#### 37.

$$0.02$$

#### 39.

$$\dfrac{1}{12}\pi$$

#### 41.

$$\dfrac{1}{3}\pi$$

#### 43.

$$\dfrac{1}{4}\pi$$

#### 45.

1. $$\dfrac{5\pi}{6}\text{,}$$ $$~\dfrac{7\pi}{6}\text{,}$$ $$~\dfrac{11\pi}{6}$$
2. $$\dfrac{3\pi}{4}\text{,}$$ $$~\dfrac{5\pi}{4}\text{,}$$ $$~\dfrac{7\pi}{4}$$
3. $$\dfrac{2\pi}{3}\text{,}$$ $$~\dfrac{4\pi}{3}\text{,}$$ $$~\dfrac{5\pi}{3}$$

#### 47.

 $$~\theta~$$ $$~~~\sin (\theta)~~~$$ $$~~~\cos (\theta)~~~$$ $$~~~\tan (\theta)~~~$$ $$\dfrac{7\pi}{6}$$ $$\dfrac{-1}{2}$$ $$\dfrac{-\sqrt{3}}{2}$$ $$\dfrac{1}{\sqrt{3}}$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{-1}{\sqrt{2}}$$ $$\dfrac{-1}{\sqrt{2}}$$ $$1$$ $$\dfrac{4\pi}{3}$$ $$\dfrac{-\sqrt{3}}{2}$$ $$\dfrac{-1}{2}$$ $$\sqrt{3}$$

#### 49.

$$\dfrac{1}{4}$$

#### 51.

$$-\dfrac{3+\sqrt{3}}{3}$$

#### 53.

$$\dfrac{3-6\sqrt{3}}{4}$$

#### 55.

$$(\cos (2.5),\sin (2.5)) \approx (-0.8, 0.6)$$

#### 57.

$$(\cos (8.5), \sin (8.5)) \approx (-0.6, 0.8)$$

#### 59.

$$\cos (5) \approx 0.28$$ mi east, $$\sin (5) \approx -0.96$$ mi north, or about 0.96 mi south

#### 61.

$$1.75$$

#### 63.

$$5.8$$

#### 65.

$$3.84$$

#### 67.

1. Intersections: $$\left(\dfrac{1}{\sqrt{2}},\dfrac{1}{\sqrt{2}}\right)$$ and $$\left(\dfrac{-1}{\sqrt{2}},\dfrac{-1}{\sqrt{2}}\right)$$
2. $$(\cos\left(\dfrac{\pi}{4}\right),\sin\left(\dfrac{\pi}{4}\right))$$ and $$\left(\cos\left(\dfrac{5\pi}{4}\right),\sin\left(\dfrac{5\pi}{4}\right)\right)$$

#### 69.

1. $$\displaystyle m = \dfrac{3}{8}$$
2. $$\displaystyle \tan^{-1}(\frac{3}{8})\approx 0.3588$$

#### 71.

$$y - 2 = \sqrt{3}(x - 4)$$

#### 73.

$$y + 8 = (\tan (2.4))((x - 5)$$ or $$y + 8 = -0.916(x - 5)$$

#### 75.

Any point $$(x,y)$$ on the terminal side of $$\theta$$ satisfies $$\cos (\theta) = \dfrac{x}{r}\text{,}$$ $$~ \sin (\theta) = \dfrac{y}{r}\text{.}$$ For the point $$P$$ where $$r = 1\text{,}$$ $$~\cos (\theta) = x\text{,}$$ $$~\sin (\theta) = y\text{.}$$ The arc of length $$t$$ is spanned by an angle $$\theta$$ in standard position. Because arclength is $$r\theta$$ and $$r = 1\text{,}$$ $$~ t = \theta,$$ so $$x = \cos (t)\text{,}$$ $$~ y = \sin (t)\text{.}$$

#### 77.

The two right triangles shown are similar, so their sides are proportional. The hypotenuse of the large triangle is $$r$$ times the hypotenuse of the small triangle, so the two legs of the large triangle must be $$r$$ times the legs of the small triangle. Thus, because the coordinates of the vertex on the unit circle are $$(\cos (\theta), \sin (\theta))\text{,}$$ the coordinates of $$P$$ must be $$(r\cos (\theta), r\sin (\theta))\text{.}$$

#### 79.

71 m west, 587 m north

### 6.3Graphs of the Circular FunctionsHomework 6.3

#### 1.

1.  $$\theta$$ $$0$$ $$\dfrac{\pi}{12}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{5\pi}{12}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{7\pi}{12}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{3\pi}{4}$$ $$\dfrac{5\pi}{6}$$ $$\dfrac{11\pi}{12}$$ $$\pi$$ $$\cos (\theta)$$ $$1$$ $$0.97$$ $$0.87$$ $$0.71$$ $$0.50$$ $$0.26$$ $$0$$ $$-0.26$$ $$-0.50$$ $$-0.71$$ $$-0.87$$ $$-0.97$$ $$-1$$

#### 5.

1. Domain: $$(-\infty, \infty)\text{,}$$ range: $$[-1,1]$$

#### 7.

1. Domain: $$x \ne \dfrac{n\pi}{2},~n$$ an odd integer, range: $$(-\infty, \infty)$$

#### 9.

1. $$x \approx 0.7$$ or $$x \approx 2.4$$
2. $$x \approx 0.36$$ or $$x \approx 2.78$$

#### 11.

1. $$x \approx 2$$ or $$x \approx 4.3$$
2. $$x \approx 2.5$$ or $$x \approx 3.79$$

#### 13.

$$x \approx 1.3$$ or $$x \approx 4.5$$

#### 15.

$$x \approx 2.7$$ or $$x \approx 5.8$$

#### 17.

$$x \approx 1.4$$ or $$x \approx 4.5$$

#### 19.

$$x \approx 2.2$$ or $$x \approx 5.3$$

#### 21.

I: 0.5, II: 2.7, III: 3.6, IV: 5.8

#### 23.

I: 0.6, II: 2.6, III: 3.7, IV: 5.7

#### 25.

I: 1.3, II: 1.8, III: 4.5, IV: 4.9

#### 27.

$$t \approx 0.74$$ or $$t \approx 5.55$$

#### 29.

$$t \approx 1.01$$ or $$t \approx 4.15$$

#### 31.

$$x \approx 3.94$$ or $$x \approx 5.48$$

#### 33.

$$t = \dfrac{3\pi}{2}$$

#### 35.

$$x = \dfrac{\pi}{4}~$$ or $$~x = \dfrac{5\pi}{4}$$

#### 37.

$$z = \dfrac{\pi}{3}~$$ or $$~z = \dfrac{5\pi}{3}$$

#### 39.

$$s = \dfrac{2\pi}{3}~$$ or $$~s = \dfrac{5\pi}{3}$$

#### 41.

$$t = \dfrac{5\pi}{4}~$$ or $$~t = \dfrac{7\pi}{4}$$

#### 43.

$$x = \dfrac{5\pi}{6}~$$ or $$~x = \dfrac{7\pi}{6}$$

#### 45.

1. $$\displaystyle 0.78$$
2. $$\displaystyle 1.12$$

#### 47.

1. $$\displaystyle 0.26$$
2. $$\displaystyle 1.28$$

#### 49.

1. $$\displaystyle -0.9$$
2. No solution

#### 51.

1. $$\displaystyle \dfrac{1}{\sqrt{2}}$$
2. $$\displaystyle 0.9$$

#### 53.

$$-6\sqrt{2}$$

#### 55.

$$-4\sqrt{3}$$

#### 57.

$$6$$

#### 59.

b-c.
d. $$t \approx 10$$ and $$t \approx 20~~~$$ e. $$t \approx 7.5$$ to $$t \approx 22$$

#### 61.

b-c.
d. High: day 204, $$105\degree\text{;}$$ low: day 25, $$66\degree$$ e. $$d \approx 128$$ to $$d \approx 281$$

#### 63.

1. $$\displaystyle -0.8,~ 0.6,~ \dfrac{-4}{3}$$
2. $$\displaystyle 0.8,~ -0.6,~ \dfrac{-4}{3}$$
3. $$\displaystyle -0.8,~ -0.6,~ \dfrac{4}{3}$$

#### 65.

1. $$\displaystyle 0.92,~ -0.39,~ \dfrac{-92}{39}$$
2. $$\displaystyle -0.92,~ 0.39,~ \dfrac{-92}{39}$$
3. $$\displaystyle 0.92,~ 0.39,~ \dfrac{92}{39}$$

#### 71.

1. Domain: $$(-\infty, \infty)\text{,}$$ range: $$(-\infty, 9]$$

#### 73.

1. Domain: $$x \ne 0\text{,}$$ range: $$(-\infty, 2)$$

#### 75.

1. Domain: $$[6, \infty)\text{,}$$ range: $$[0, \infty)$$

#### 77.

1. Domain: $$[-2,2]\text{,}$$ range: $$[-2,0]$$

#### 79.

1.  $$x$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$\cos (x)$$ $$1$$ $$0$$ $$-1$$ $$0$$ $$1$$
2. Domain: $$(-\infty, \infty) \text{,}$$ Range: $$[-1,1]$$

### 6.4Chapter 6 Summary and ReviewChapter 6 Review Problems

#### 1.

1. $$\displaystyle \dfrac{5\pi}{12}$$
2. $$\displaystyle \dfrac{7\pi}{6}$$
3. $$\displaystyle \dfrac{17\pi}{9}$$

#### 3.

1. $$\displaystyle 0.47$$
2. $$\displaystyle 2.48$$
3. $$\displaystyle 3.80$$

#### 5.

1. $$\displaystyle 150\degree$$
2. $$\displaystyle 54\degree$$
3. $$\displaystyle 230\degree$$

#### 7.

1. $$\displaystyle 114.59\degree$$
2. $$\displaystyle 206.26\degree$$
3. $$\displaystyle 45.84\degree$$

#### 9.

1. $$\displaystyle \dfrac{4\pi}{3}$$
2. $$\displaystyle \dfrac{7\pi}{6}$$
3. $$\displaystyle \dfrac{9\pi}{4}$$

#### 11.

1. $$\displaystyle \dfrac{1}{8}$$
2. $$\displaystyle \dfrac{5}{16}$$
3. $$\displaystyle \dfrac{7}{6}$$

1. II
2. I
3. IV

#### 15.

1. $$\displaystyle 0.006,~2.17,~0.0379$$
2. $$\displaystyle 0.0379$$

#### 17.

$$6885$$ mph

#### 19.

1. $$\displaystyle 0$$
2. $$\displaystyle \dfrac{-8}{\sqrt{3}}$$
3. $$\displaystyle \dfrac{-1}{2}$$

#### 21.

1. $$\displaystyle (0.5, 0.8)$$
2. $$\displaystyle (-0.4,0.9)$$
3. $$\displaystyle (-1.0,0.1)$$

#### 23.

1. $$\displaystyle (r \cos (\alpha), r \sin (\alpha))$$
2. $$\displaystyle (-r \cos (\alpha), r \sin (\alpha))$$
3. $$\displaystyle (-r \cos (\alpha), -r \sin (\alpha))$$
4. $$\displaystyle (r \cos (\alpha), -r \sin (\alpha))$$

#### 25.

$$6\pi$$

#### 27.

$$\gt$$

#### 29.

$$\lt$$

#### 31.

$$9.86$$

#### 33.

$$-1.33$$

#### 35.

1. $$\displaystyle \dfrac{\pi}{6}$$
2. $$\displaystyle \dfrac{\pi}{4}$$
3. $$\displaystyle \dfrac{3\pi}{8}$$
4. $$\displaystyle \dfrac{5\pi}{12}$$

#### 37.

1. $$\displaystyle 0.34$$
2. $$\displaystyle 0.76$$
3. $$\displaystyle 1.25$$
4. $$\displaystyle 1.5$$

#### 39.

$$158.2\degree$$

#### 43.

1. mid: $$y = 5\text{,}$$ amp: $$3\text{,}$$ period: $$\pi$$
2. $$0.86,~2.28,~4.00,~5.42$$

#### 45.

1. mid: $$y = 10\text{,}$$ amp: $$4.8\text{,}$$ period: $$2\pi$$
2. $$1.93,~4.2$$

#### 47.

$$\dfrac{5\pi}{12},~\dfrac{17\pi}{12}$$

#### 49.

$$\dfrac{\pi}{3},~\dfrac{2\pi}{3}$$

#### 51.

$$\pi$$

#### 53.

$$1.37,~4.51$$

#### 55.

$$6.02,~3.40$$

#### 57.

$$0.32,~5.97$$

#### 59.

1. $$\displaystyle 1.21,~5.07$$
2. $$\displaystyle 0.9394$$

#### 61.

1. $$\displaystyle 0.40,~2.74$$
2. $$\displaystyle 0.3827$$

#### 63.

Dom: all real numbers, Rge: $$y \ge 4$$

#### 65.

Dom: $$-4 \le s \le 4\text{,}$$ Rge: $$-4 \le y \le 0$$

#### 67.

1. $$\displaystyle x^2 + y^2 = 1$$
2. $$\displaystyle (\cos (t), \sin (t))$$
3. $$\displaystyle \cos^2 (t) + \sin^2 (t) = 1$$
4. Yes

### 7Circular Functions7.1Transformations of GraphsHomework 7-1

#### 1.

amplitude $$2\text{,}$$ period $$2\pi\text{,}$$ midline $$y=-3$$

#### 3.

amplitude $$1\text{,}$$ period $$\dfrac{\pi}{2}\text{,}$$ midline $$y=0$$

#### 5.

amplitude $$5\text{,}$$ period $$6\pi\text{,}$$ midline $$y=0$$

#### 7.

amplitude $$1\text{,}$$ period $$2\text{,}$$ midline $$y=1$$

#### 17.

$$y=-2\sin (x)$$

#### 19.

$$y=-2\cos (x)$$

#### 21.

$$y=-0.75\cos (x)$$

#### 23.

1. amplitude $$2\text{,}$$ period $$\dfrac{2\pi}{3}\text{,}$$ midline $$y=0$$
2. $$\displaystyle y=-2\sin (3x)$$

#### 25.

1. amplitude $$3\text{,}$$ period $$2\pi\text{,}$$ midline $$y=0$$
2. $$\displaystyle y=3\sin \left(\dfrac{x}{2}\right)$$

#### 27.

1. amplitude $$0.5\text{,}$$ period $$4\pi\text{,}$$ midline $$y=3.5$$
2. $$\displaystyle y=0.5\cos \left(\dfrac{x}{2}\right)+3.5$$

#### 29.

1. amplitude $$2\text{,}$$ period $$4\text{,}$$ midline $$y=-1$$
2. $$\displaystyle y=-1+2\sin \left(\dfrac{\pi x}{2}\right)$$

#### 31.

1.  $$t$$ $$2t$$ $$\cos (2t)$$ $$-5\cos (2t)$$ $$2-5\cos (2t)$$ $$0$$ $$0$$ $$1$$ $$-5$$ $$-3$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$0$$ $$0$$ $$2$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$-1$$ $$5$$ $$7$$ $$\dfrac{3\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$0$$ $$0$$ $$2$$ $$\pi$$ $$2\pi$$ $$1$$ $$-5$$ $$-3$$

#### 33.

1.  $$t$$ $$\dfrac{t}{2}$$ $$\cos \left(\dfrac{t}{2}\right)$$ $$3\cos \left(\dfrac{t}{2}\right)$$ $$1+3\cos \left(\dfrac{t}{2}\right)$$ $$0$$ $$0$$ $$1$$ $$3$$ $$4$$ $$\pi$$ $$\dfrac{\pi}{2}$$ $$0$$ $$0$$ $$1$$ $$2\pi$$ $$\pi$$ $$-1$$ $$-3$$ $$-2$$ $$3\pi$$ $$\dfrac{3\pi}{2}$$ $$0$$ $$0$$ $$1$$ $$4\pi$$ $$2\pi$$ $$1$$ $$3$$ $$4$$

#### 35.

1.  $$t$$ $$\dfrac{t}{3}$$ $$\sin \left(\dfrac{t}{3}\right)$$ $$2\sin \left(\dfrac{t}{3}\right)$$ $$-3+2\sin \left(\dfrac{t}{3}\right)$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-3$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{\pi}{2}$$ $$1$$ $$2$$ $$-1$$ $$3\pi$$ $$\pi$$ $$0$$ $$0$$ $$-3$$ $$\dfrac{9\pi}{2}$$ $$\dfrac{3\pi}{2}$$ $$-1$$ $$-2$$ $$-5$$ $$6\pi$$ $$2\pi$$ $$0$$ $$0$$ $$-3$$

#### 45.

1. $$\displaystyle W(t)=12+8\cos \left(\dfrac{\pi t}{6}\right)$$

#### 47.

1. $$\displaystyle h=10+14\cos \left(\dfrac{\pi t}{5}\right)$$

#### 49.

$$H=12-2.4\cos \left(\dfrac{\pi t}{6}\right)$$

#### 51.

$$y=155\cos(120 \pi t)$$

#### 53.

1.  $$x$$ $$\dfrac{-\pi}{4}$$ $$\dfrac{-\pi}{8}$$ $$0$$ $$\dfrac{\pi}{8}$$ $$\dfrac{\pi}{4}$$ $$\tan 2x$$ undef $$-1$$ $$0$$ $$1$$ undef
2. period $$\dfrac{\pi}{2}\text{,}$$ midline $$y=0$$

#### 55.

1.  $$x$$ $$\dfrac{-\pi}{6}$$ $$\dfrac{-\pi}{12}$$ $$0$$ $$\dfrac{\pi}{12}$$ $$\dfrac{\pi}{6}$$ $$4+2\tan 3x$$ undef $$2$$ $$0$$ $$6$$ undef
2. period $$\dfrac{\pi}{3}\text{,}$$ midline $$y=4$$

#### 57.

1.  $$x$$ $$-2\pi$$ $$-\pi$$ $$0$$ $$\pi$$ $$2\pi$$ $$3-\tan \left(\dfrac{x}{4}\right)$$ undef $$4$$ $$0$$ $$2$$ undef
2. period $$4\pi\text{,}$$ midline $$y=3$$

#### 59.

$$\dfrac{\pi}{12}\text{,}$$ $$~\dfrac{5\pi}{12}\text{,}$$ $$~\dfrac{7\pi}{12}\text{,}$$ $$~\dfrac{11\pi}{12}\text{,}$$ $$~\dfrac{13\pi}{12}\text{,}$$ $$~\dfrac{17\pi}{12}\text{,}$$ $$~\dfrac{19\pi}{12}\text{,}$$ $$~\dfrac{23\pi}{12}$$

#### 61.

$$\dfrac{7\pi}{12}\text{,}$$ $$~\dfrac{11\pi}{12}\text{,}$$ $$~\dfrac{19\pi}{12}\text{,}$$ $$~\dfrac{23\pi}{12}$$

#### 63.

$$\dfrac{\pi}{12}\text{,}$$ $$~\dfrac{5\pi}{12}\text{,}$$ $$~\dfrac{3\pi}{4}\text{,}$$ $$~\dfrac{13\pi}{12}\text{,}$$ $$~\dfrac{17\pi}{12}\text{,}$$ $$~\dfrac{7\pi}{4}$$

#### 65.

$$1.83,~2.88,~4.97,~6.02$$

#### 67.

$$4.19$$

#### 69.

$$0.28,~1.81,~2.37,~3.91,~4.47,~6.00$$

### 7.2The General Sinusoidal FunctionHomework 7-2

#### 1.

1.  $$x$$ $$-\pi$$ $$\dfrac{-5\pi}{6}$$ $$\dfrac{-2\pi}{3}$$ $$\dfrac{-\pi}{2}$$ $$\dfrac{-\pi}{3}$$ $$\dfrac{-\pi}{6}$$ $$0$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{5\pi}{6}$$ $$\pi$$ $$f(x)$$ $$0$$ $$\dfrac{-1}{2}$$ $$\dfrac{-\sqrt{3}}{2}$$ $$-1$$ $$\dfrac{-\sqrt{3}}{2}$$ $$\dfrac{-1}{2}$$ $$0$$ $$\dfrac{1}{2}$$ $$\dfrac{\sqrt{3}}{2}$$ $$1$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{1}{2}$$ $$0$$ $$g(x)$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{1}{2}$$ $$0$$ $$\dfrac{-1}{2}$$ $$\dfrac{-\sqrt{3}}{2}$$ $$-1$$ $$\dfrac{-\sqrt{3}}{2}$$ $$\dfrac{-1}{2}$$ $$0$$ $$\dfrac{1}{2}$$ $$\dfrac{\sqrt{3}}{2}$$ $$1$$ $$\dfrac{\sqrt{3}}{2}$$
2. $$\dfrac{\pi}{3}$$ to the right
3. $$\displaystyle \dfrac{5\pi}{6}$$
4. $$\displaystyle \dfrac{-2\pi}{3},~\dfrac{\pi}{3}$$

#### 3.

1.  $$x$$ $$-\pi$$ $$\dfrac{-3\pi}{4}$$ $$\dfrac{-\pi}{2}$$ $$\dfrac{-\pi}{4}$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$f(x)$$ $$0$$ $$1$$ undef $$-1$$ $$0$$ $$1$$ undef $$-1$$ $$0$$ $$g(x)$$ $$1$$ undef $$-1$$ $$0$$ $$1$$ undef $$-1$$ $$0$$ $$1$$
2. $$\dfrac{\pi}{4}$$ to the left
3. $$\displaystyle -\pi,~ 0,~ \pi$$
4. $$\displaystyle \dfrac{-\pi}{4},~\dfrac{-3\pi}{4}$$

#### 5.

1. amplitude 2, shift $$\dfrac{\pi}{6}$$ to the left
2.  $$x$$ $$x+\dfrac{\pi}{6}$$ $$\cos\left(x+\dfrac{\pi}{6}\right)$$ $$-2\cos\left(x+\dfrac{\pi}{6}\right)$$ $$\dfrac{-7\pi}{6}$$ $$-\pi$$ $$-1$$ $$2$$ $$\dfrac{-2\pi}{3}$$ $$\dfrac{-\pi}{2}$$ $$0$$ $$0$$ $$\dfrac{-\pi}{6}$$ $$0$$ $$1$$ $$-2$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{2}$$ $$0$$ $$0$$ $$\dfrac{5\pi}{6}$$ $$\pi$$ $$-1$$ $$2$$ $$\dfrac{4\pi}{3}$$ $$\dfrac{3\pi}{2}$$ $$0$$ $$0$$ $$\dfrac{11\pi}{6}$$ $$2\pi$$ $$1$$ $$-2$$
3. $$\displaystyle \dfrac{\pi}{2},~ \dfrac{7\pi}{6}$$
4. $$\displaystyle \dfrac{\pi}{3},~ \dfrac{4\pi}{3}$$

#### 7.

1. $$\displaystyle f(x)=\sin \left(x+\dfrac{\pi}{4}\right)$$
2. $$\displaystyle f(x)=\cos \left(x-\dfrac{\pi}{4}\right)$$

#### 9.

1. $$\displaystyle f(x)=\tan \left(x-\dfrac{\pi}{3}\right)$$
2. $$\displaystyle f(x)=\tan \left(x+\dfrac{2\pi}{3}\right)$$

#### 11.

1. period $$\pi\text{,}$$ shift $$\dfrac{\pi}{6}$$ to the right
2.  $$x$$ $$2x$$ $$2x-\dfrac{\pi}{3}$$ $$\cos\left(2x-\dfrac{\pi}{3}\right)$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{3}$$ $$0$$ $$1$$ $$\dfrac{5\pi}{12}$$ $$\dfrac{5\pi}{6}$$ $$\dfrac{\pi}{2}$$ $$0$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{4\pi}{3}$$ $$\pi$$ $$-1$$ $$\dfrac{11\pi}{12}$$ $$\dfrac{11\pi}{6}$$ $$\dfrac{3\pi}{2}$$ $$0$$ $$\dfrac{7\pi}{6}$$ $$\dfrac{7\pi}{3}$$ $$2\pi$$ $$1$$
3. $$\displaystyle \dfrac{\pi}{6},~ \dfrac{7\pi}{6}$$
4. $$\displaystyle \dfrac{5\pi}{12},~ \dfrac{11\pi}{12},~ \dfrac{13\pi}{6},~ \dfrac{23\pi}{12}$$

#### 13.

1. period 2, shift $$\dfrac{1}{3}$$ to the left
2.  $$x$$ $$\pi x$$ $$\pi x+\dfrac{\pi}{3}$$ $$\sin\left(\pi x+\dfrac{\pi}{3}\right)$$ $$\dfrac{-1}{3}$$ $$\dfrac{-\pi}{3}$$ $$0$$ $$0$$ $$\dfrac{1}{6}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{2}$$ $$1$$ $$\dfrac{2}{3}$$ $$\dfrac{2\pi}{3}$$ $$\pi$$ $$0$$ $$\dfrac{7}{6}$$ $$\dfrac{7\pi}{6}$$ $$\dfrac{3\pi}{2}$$ $$-1$$ $$\dfrac{5}{3}$$ $$\dfrac{5\pi}{3}$$ $$2\pi$$ $$0$$
3. $$\displaystyle \dfrac{-11}{6},~ \dfrac{1}{6}$$
4. $$\displaystyle \dfrac{-4}{3},~ \dfrac{-1}{3},~ \dfrac{2}{3},~ \dfrac{5}{3}$$

#### 15.

1. midline $$y=4\text{,}$$ period $$4\pi\text{,}$$ horizontal shift $$\dfrac{\pi}{3}$$ to the right, amplitude 3
2.  $$x$$ $$\dfrac{x}{2}$$ $$\dfrac{x}{2}-\dfrac{\pi}{6}$$ $$\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)$$ $$3\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)+4$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{6}$$ $$0$$ $$0$$ $$4$$ $$\dfrac{4\pi}{3}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{\pi}{2}$$ $$1$$ $$7$$ $$\dfrac{7\pi}{3}$$ $$\dfrac{7\pi}{6}$$ $$\pi$$ $$0$$ $$3$$ $$\dfrac{10\pi}{3}$$ $$\dfrac{5\pi}{3}$$ $$\dfrac{3\pi}{2}$$ $$-1$$ $$1$$ $$\dfrac{13\pi}{3}$$ $$\dfrac{13\pi}{6}$$ $$2\pi$$ $$0$$ $$4$$
3. no solution for $$0 \le x \le 2\pi$$
4. $$\displaystyle \dfrac{\pi}{3}$$

#### 17.

$$y=2\sin\left(\dfrac{2\pi}{3}(x+4)\right)+5$$

#### 19.

$$y=-5\cos\left(\dfrac{\pi x}{180}\right)+12$$

#### 21.

1. $$\displaystyle f(x)=3\sin \left(x+\dfrac{2\pi}{3}\right)$$
2. $$\displaystyle f(x)=3\cos \left(x+\dfrac{\pi}{6}\right)$$

#### 23.

1. $$\displaystyle f(x)=2\sin (2(x-\dfrac{\pi}{4}))$$
2. $$\displaystyle f(x)=-2\cos (2x)$$

#### 25.

1. $$\displaystyle f(x)=4\sin \left[\dfrac{1}{4}\left(x-\dfrac{7\pi}{3}\right)\right]$$
2. $$\displaystyle f(x)=-4\cos \left[\dfrac{1}{4}\left(x-\dfrac{\pi}{3}\right)\right]$$

#### 27.

1. midline $$T=35.35\text{,}$$ period 12, amplitude 36.95
2. $$\displaystyle T(m)=-36.95 \cos \left(\dfrac{\pi}{6} m\right)+35.35$$

#### 29.

1. midline $$h=1.4\text{,}$$ period $$\dfrac{2 \pi}{0.51} \approx{12.32}\text{,}$$ amplitude 1.4
2. high 11:10 am, low 5:19 pm

#### 31.

1. amplitude 3.2, period 2, midline $$y=2$$
2. $$\displaystyle f(t)=2+3.2\cos (\pi t)$$

#### 33.

1. amplitude 5, period 1, midline $$y=0$$
2. $$\displaystyle H(x)=5\sin (2\pi x) + 5$$

### 7.3Solving EquationsHomework 7-3

#### 1.

$$\dfrac{3\pi}{8}\text{,}$$ $$~ \dfrac{7\pi}{8}\text{,}$$ $$~ \dfrac{11\pi}{8}\text{,}$$ $$~ \dfrac{15\pi}{8}$$

#### 3.

$$0\text{,}$$ $$~ \dfrac{\pi}{2}\text{,}$$ $$~ \pi\text{,}$$ $$~ \dfrac{3\pi}{2}\text{,}$$ $$~ 2\pi$$

#### 5.

$$\dfrac{2\pi}{9}\text{,}$$ $$~ \dfrac{4\pi}{9}\text{,}$$ $$~ \dfrac{8\pi}{9}\text{,}$$ $$~ \dfrac{10\pi}{9}\text{,}$$ $$~ \dfrac{14\pi}{9}\text{,}$$ $$~ \dfrac{16\pi}{9}$$

#### 7.

$$\dfrac{\pi}{12}\text{,}$$ $$~ \dfrac{5\pi}{12}\text{,}$$ $$~ \dfrac{13\pi}{12}\text{,}$$ $$~ \dfrac{17\pi}{12}$$

#### 9.

$$\dfrac{\pi}{18}\text{,}$$ $$~ \dfrac{7\pi}{18}\text{,}$$ $$~ \dfrac{13\pi}{18}\text{,}$$ $$~ \dfrac{19\pi}{18}\text{,}$$ $$~ \dfrac{25\pi}{18}\text{,}$$ $$~ \dfrac{31\pi}{18}$$

#### 11.

$$0.491,~ 2.651,~ 3.632,~ 5.792$$

#### 13.

$$0.540\text{,}$$ $$~ 1.325\text{,}$$ $$~ 2.110\text{,}$$ $$~ 2.896\text{,}$$ $$~ 3.681\text{,}$$ $$~ 4.467\text{,}$$ $$~ 5.252\text{,}$$ $$~ 6.037$$

#### 15.

$$1.114\text{,}$$ $$~ 2.027\text{,}$$ $$~ 3.209\text{,}$$ $$~ 4.122\text{,}$$ $$~ 5.303\text{,}$$ $$~ 6.216$$

#### 17.

$$0.702\text{,}$$ $$~ 2.440\text{,}$$ $$~ 3.843\text{,}$$ $$~ 5.582$$

#### 19.

$$0\text{,}$$ $$~ 1\text{,}$$ $$~ 2\text{,}$$ $$~ 3\text{,}$$ $$~ 4\text{,}$$ $$~ 5\text{,}$$ $$~ 6$$

#### 21.

$$\dfrac{\pi}{6}\text{,}$$ $$~ \dfrac{2\pi}{3}\text{,}$$ $$~ \dfrac{7\pi}{6}\text{,}$$ $$~ \dfrac{5\pi}{3}$$

#### 23.

$$\dfrac{5\pi}{12}\text{,}$$ $$~ \dfrac{7\pi}{12}\text{,}$$ $$~ \dfrac{13\pi}{12}\text{,}$$ $$~ \dfrac{5\pi}{4}\text{,}$$ $$~ \dfrac{7\pi}{4}\text{,}$$ $$~ \dfrac{23\pi}{12}$$

#### 25.

$$\dfrac{3\pi}{2}$$

#### 27.

$$\dfrac{7}{6}\text{,}$$ $$~ \dfrac{11}{6}\text{,}$$ $$~ \dfrac{19}{6}\text{,}$$ $$~ \dfrac{23}{6}\text{,}$$ $$~ \dfrac{31}{6}\text{,}$$ $$~ \dfrac{35}{6}$$

#### 29.

$$1.14\text{,}$$ $$~ 1.62\text{,}$$ $$~ 3.23\text{,}$$ $$~ 3.72\text{,}$$ $$~ 5.24\text{,}$$ $$~ 5.81$$

#### 31.

$$0.44\text{,}$$ $$~ 1.44\text{,}$$ $$~ 2.44\text{,}$$ $$~ 3.44\text{,}$$ $$~ 4.44\text{,}$$ $$~ 5.44$$

#### 33.

$$0.01\text{,}$$ $$~ 3.39\text{,}$$ $$~ 6.01$$

#### 35.

$$0.564\text{,}$$ $$~ 1.182\text{,}$$ $$~ 2.658\text{,}$$ $$~ 3.276\text{,}$$ $$~ 4.752\text{,}$$ $$~ 5.371$$

#### 37.

$$0.423\text{,}$$ $$~ 2.977\text{,}$$ $$~ 4.423$$

#### 39.

$$1.165,~ 4.165$$

#### 41.

$$2.251$$

#### 43.

1. $$\displaystyle P(t)=4000\cos\left(\dfrac{\pi}{6}t\right)+46,000$$
2. $$t=\cos^{-1}\left( \dfrac{-1}{4} \right)\cdot\dfrac{6}{\pi}\approx3.48$$ months (Dec) or $$t=12-\cos^{-1}\left( \frac{-1}{4} \right)\cdot\frac{6}{\pi}\approx 8.52$$ months (June)
3. $$P(t)$$ is less than 45,000 between $$A$$ and $$B\text{.}$$

#### 45.

1. $$\displaystyle h(t)=11-10\cos\left(\dfrac{\pi}{30}t\right)$$
2. $$t=\cos\left(-0.7 \right)\cdot \dfrac{30}{\pi}\approx 22.40$$ sec or $$t=60-\cos\left(-0.7 \right)\cdot \dfrac{30}{\pi}\approx 37.60$$ sec
3. Delbert is above 18 m between $$A$$ and $$B\text{.}$$

### 7.4Chapter 7 Summary and ReviewReview Problems

#### 1.

amp: $$2\text{,}$$ period: $$\dfrac{2\pi}{3} \text{;}$$ mid: $$y=4$$

#### 3.

amp: $$2.5\text{,}$$ period: $$2\text{;}$$ mid: $$y=-2$$

#### 9.

$$y=3+2\sin (x)$$

#### 11.

$$y=4-3\sin \left(\dfrac{x}{4}\right)$$

#### 13.

1. period: $$4\pi\text{,}$$ shift: $$\dfrac{\pi}{3}$$ left
2.  $$x$$ $$\dfrac{x}{2}$$ $$\dfrac{x}{2}+\dfrac{\pi}{6}$$ $$\sin\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)$$ $$\dfrac{-2\pi}{3}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{-\pi}{6}$$ $$\dfrac{-1}{2}$$ $$\dfrac{-\pi}{3}$$ $$\dfrac{-\pi}{6}$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\dfrac{\pi}{6}$$ $$\dfrac{1}{2}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{12}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{1}{\sqrt{2}}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\sqrt{3}}{2}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{2}$$ $$1$$ $$\pi$$ $$\dfrac{\pi}{2}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{\sqrt{3}}{2}$$
3. $$\displaystyle \dfrac{2\pi}{3}$$
4. $$\displaystyle \dfrac{-\pi}{3}$$

#### 15.

1. mid: $$y=20\text{,}$$ period: 0, amp: 5
2. Fill in the table of values.
 $$x$$ $$\dfrac{\pi}{30}x$$ $$\cos\left(\dfrac{\pi}{30}x\right)$$ $$20-5\cos\left(\dfrac{\pi}{30}x\right)$$ $$-5$$ $$\dfrac{-\pi}{6}$$ $$\dfrac{\sqrt{3}}{2}$$ $$20-\dfrac{\sqrt{3}}{2}$$ $$0$$ $$0$$ $$1$$ $$15$$ $$5$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\sqrt{3}}{2}$$ $$20-\dfrac{\sqrt{3}}{2}$$ $$10$$ $$\dfrac{\pi}{3}$$ $$\dfrac{1}{2}$$ $$17.5$$ $$15$$ $$\dfrac{\pi}{2}$$ $$0$$ $$20$$ $$50$$ $$\pi$$ $$-1$$ $$25$$
3. 30
4. 15, 45

#### 19.

1. 0.57, 3.07, 3.71

#### 21.

$$y=85.5-19.5\cos\left(\dfrac{\pi}{6}t\right)$$

#### 23.

1. amp: 3, period: 12, midline: $$y=15$$
2. $$\displaystyle y=15-3\cos\left(\frac{\pi}{6}t\right)$$

#### 25.

$$\dfrac{7\pi}{12}\text{,}$$ $$\dfrac{11\pi}{12}\text{,}$$ $$\dfrac{19\pi}{12}\text{,}$$ $$\dfrac{23\pi}{12}$$

#### 27.

$$0\text{,}$$ $$\dfrac{\pi}{4}\text{,}$$ $$\dfrac{\pi}{2}\text{,}$$ $$\dfrac{3\pi}{4}\text{,}$$ $$\pi\text{,}$$ $$\dfrac{5\pi}{4}\text{,}$$ $$\dfrac{7\pi}{4}\text{,}$$ $$2\pi$$

#### 29.

0.066, 1.113, 2.160, 3.207, 4.255, 5.302

#### 31.

1.150, 1.991, 4.292, 5.133

#### 33.

$$\dfrac{\pi}{24} \text{,}$$ $$\dfrac{5\pi}{24} \text{,}$$ $$\dfrac{25\pi}{24} \text{,}$$ $$\dfrac{29\pi}{24}$$

No solution

#### 37.

0.375, 1.422, 2.470, 3.517, 4.564, 5.611

2.120, 4.880

### 8More Functions and Identities8.1Sum and Difference FormulasHomework 8-1

#### 1.

$$x_2=x_1\text{,}$$ $$y_2=-y_1\text{,}$$ and $$r_2=r_1\text{.}$$ Thus, $$\cos(-\alpha)=\dfrac{x_2}{r_2} =\dfrac{x_1}{r_1}=\cos(\alpha) \text{,}$$ $$\sin(-\alpha)=\dfrac{y_2}{r_2} =\dfrac{-y_1}{r_1}=-\sin(\alpha) \text{,}$$ and $$\tan(-\alpha)=\dfrac{y_2}{x_2} =\dfrac{-y_1}{x_1}=-\tan(\alpha) \text{.}$$

#### 3.

$$\dfrac{-(\sqrt{2} + \sqrt{6})}{4}$$

#### 5.

$$\cos(0.3-2x)=0.24\text{,}$$ $$\sin(0.3-2x)=0.97$$

#### 7.

$$\cos(45\degree + 45\degree)=\cos(90\degree)=0\text{,}$$ but $$\cos (45\degree) +\cos (45\degree) = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}$$

#### 9.

$$\tan(87\degree -29\degree)\approx 1.600\text{,}$$ but $$\tan (87\degree) -\tan (29\degree) \approx 18.527$$

#### 11.

The curves are different.

#### 13.

1. $$\displaystyle \dfrac{63}{65}$$
2. $$\displaystyle \dfrac{-16}{65}$$
3. $$\displaystyle \dfrac{-16}{63}$$

#### 15.

1. $$\displaystyle \dfrac{44}{117}$$
2. $$\displaystyle \dfrac{4}{3}$$

#### 17.

1. $$\displaystyle \dfrac{36}{85}$$
2. $$\displaystyle \dfrac{-13}{84}$$

#### 19.

1. $$\displaystyle \dfrac{-16}{65}$$
2. $$\displaystyle \dfrac{63}{65}$$
3. $$\displaystyle \dfrac{-16}{63}$$

#### 21.

$$\cos (15\degree)=\dfrac{\sqrt{6}+\sqrt{2}}{4} \text{,}$$ $$\tan (15\degree) = 2-\sqrt{3}$$

#### 23.

$$\dfrac{6\sqrt{2}+1}{10}$$

#### 25.

$$\cos (\theta)$$

#### 27.

$$\dfrac{\sqrt{3}}{2}\cos (t) -\dfrac{1}{2}\sin (t)$$

#### 29.

$$\dfrac{\sqrt{3}\tan\beta -1}{\sqrt{3}+\tan\beta}$$

No

No

#### 35.

$$1=2\left(\dfrac{1}{\sqrt{2}} \right)\left(\dfrac{1}{\sqrt{2}} \right)$$

#### 37.

$$\frac{1}{2} =\left(\dfrac{\sqrt{3}}{2} \right)^2 - \left(\dfrac{1}{2} \right)^2$$

#### 39.

False, but $$\cos (2\alpha)=2(0.32)^2-1$$

#### 41.

False, but $$2\theta = \sin^{-1}(h)$$

#### 43.

$$\sin (68\degree)$$

#### 45.

$$\cos\left(\dfrac{\pi}{8}\right)$$

#### 47.

$$\cos (6\theta)$$

#### 49.

$$\sin 10t$$

#### 51.

$$\tan 128\degree$$

#### 53.

$$\cos (4\beta)$$

#### 55.

1. $$\displaystyle \dfrac{5}{6}$$
2. $$\displaystyle \dfrac{\sqrt{11}}{6}$$
3. $$\displaystyle \dfrac{5}{\sqrt{11}}$$
4. $$\displaystyle \dfrac{5\sqrt{11}}{18}$$
5. $$\displaystyle \dfrac{-7}{18}$$
6. $$\displaystyle \dfrac{-5\sqrt{11}}{7}$$

#### 57.

1. $$\displaystyle \dfrac{1}{\sqrt{w^2+1}}$$
2. $$\displaystyle \dfrac{w}{\sqrt{w^2+1}}$$
3. $$\displaystyle \dfrac{1}{w}$$
4. $$\displaystyle \dfrac{2w}{w^2+1}$$
5. $$\displaystyle \dfrac{w^2 -1}{w^2+1}$$
6. $$\displaystyle \dfrac{2w}{w^2-1}$$

#### 59.

1. $$\displaystyle \dfrac{-5}{13}$$
2. $$\displaystyle \dfrac{-120}{169}$$
3. $$\displaystyle \dfrac{119}{169}$$
4. $$\displaystyle \dfrac{-120}{119}$$

#### 61.

1. $$\displaystyle \dfrac{8}{15}$$
2. $$\displaystyle \dfrac{-15}{17}$$
3. $$\displaystyle \dfrac{-8}{17}$$

#### 63.

1. $$\displaystyle 2\sin(\theta) \cdot \cos(\theta) +\sqrt{2}\cos(\theta)=0$$
2. $$\dfrac{\pi}{2} \text{,}$$ $$\dfrac{5\pi}{4} \text{,}$$ $$\dfrac{3\pi}{2} \text{,}$$ $$\dfrac{7\pi}{4}$$

#### 65.

1. $$\displaystyle 2\cos^2 (t) -5\cos (t) +2=0$$
2. $$\dfrac{\pi}{3} \text{,}$$ $$\dfrac{5\pi}{3}$$

#### 67.

1. $$\displaystyle \frac{2\tan(\beta)}{1-\tan^2(\beta)}+2\sin(\beta)=0$$
2. $$0\text{,}$$ $$\dfrac{\pi}{3}\text{,}$$ $$\pi$$ , $$\dfrac{5\pi}{3}$$

#### 69.

1. $$\displaystyle 3\cos(\phi) - \cos (\phi)=\sqrt{3}$$
2. $$\dfrac{\pi}{6}\text{,}$$ $$\dfrac{11\pi}{6}$$

#### 71.

1. $$\displaystyle \sin (3\phi) =1$$
2. $$\dfrac{\pi}{6}$$ , $$\dfrac{5\pi}{6}$$ , $$\dfrac{3\pi}{2}$$

#### 73.

1. $$\displaystyle \cos (\theta + 90\degree)=-\sin\theta$$
2. $$\displaystyle \sin (\theta + 90\degree)=\cos\theta$$

#### 75.

1. $$\displaystyle \cos \left(\dfrac{\pi}{2} -\theta\right) = \cos\frac{\pi}{2} \cos(\theta) + \sin \frac{\pi}{2}\sin\theta = \sin (\theta)$$
2. $$\displaystyle \sin \left(\dfrac{\pi}{2} -\theta\right) = \sin\frac{\pi}{2} \cos(\theta) - \cos \frac{\pi}{2}\sin(\theta)= \cos (\theta)$$

#### 77.

\begin{aligned}[t]\sin(2\theta) \amp=\sin(\theta + \theta)\\ \amp= \sin(\theta)\cos(\theta) + \sin(\theta)\cos(\theta) \\ \amp= 2\sin(\theta)\cos(\theta) \end{aligned}

#### 79.

1. Not an identity.
2. $$\beta=\pi$$ (many answers possible)

Identity

#### 83.

1. Not an identity.
2. $$\theta=0$$ (many answers possible)

Identity

Identity

#### 89.

1. $$\displaystyle l_1=\sin(\alpha), \, l_2=\cos(\alpha)$$
2. $$\theta_1$$ and $$\beta$$ are both complements of $$\phi\text{;}$$ $$\theta_2$$ and $$\alpha+\beta$$ are alternate interior angles
3. $$s_1=\cos(\alpha+\beta) \text{,}$$ $$s_2=\sin(\alpha+\beta)$$
4. $$s_3=\sin(\alpha)\sin(\beta) \text{,}$$ $$s_4=\sin(\alpha)\cos(\beta)$$
5. $$s_5=\cos(\alpha)\cos(\beta) \text{,}$$ $$s_6=\cos(\alpha)\sin(\beta)$$
6. $$\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) +\cos(\alpha)\sin(\beta) \text{,}$$ $$\cos(\alpha+\beta) = \cos(\alpha)\cos(\beta) +\sin(\alpha)\sin(\beta)$$

#### 91.

1. $$\displaystyle (AB)^2=2-2\cos(\alpha-\beta)$$
2. $$\displaystyle (AB)^2=(\cos(\alpha)-\cos(\beta))^2 + (\sin(\alpha) - \sin(\beta))^2$$
3. \displaystyle \begin{aligned}[t] 2-2\cos(\alpha-\beta)\amp = (\cos(\alpha)-\cos(\beta))^2 + (\sin(\alpha) - \sin(\beta))^2 \\ 2-2\cos(\alpha-\beta)\amp = \cos^2(\alpha) -2\cos(\alpha)\cos(\beta) + \cos^2 (\beta) + \,\\ \amp\hphantom{000000000} +\sin^2 (\alpha) - 2\sin(\alpha)\sin(\beta) + \sin^2(\beta) \\ 2-2\cos(\alpha-\beta)\amp = 1+1 - 2(\cos(\alpha) \cos(\beta) - \sin(\alpha)\sin(\beta)) \\ -2\cos(\alpha-\beta)\amp = -2(\cos(\alpha) \cos(\beta) - \sin(\alpha)\sin(\beta)) \\ \cos(\alpha-\beta)\amp = \cos(\alpha) \cos(\beta) - \sin(\alpha)\sin(\beta)) \end{aligned}

### 8.2Inverse Trigonometric FunctionsHomework 8-2

#### 1.

No inverse: Some horizontal lines intersect the curve in more than one point.

#### 3.

Inverse exists: The function is 1-1.

No inverse

No inverse

#### 9.

$$16.5\degree$$

#### 11.

$$46.4\degree$$

#### 13.

$$=51.9\degree$$

#### 15.

$$\dfrac{3\pi}{4}$$

#### 17.

$$\dfrac{-\pi}{6}$$

#### 19.

$$\dfrac{\pi}{6}$$

#### 21.

1. $$\displaystyle h=500 \tan(\theta)$$
2. $$\displaystyle \theta=\tan^{-1}\left(\dfrac{h}{500} \right)$$
3. $$\theta=\tan^{-1}(2) \text{,}$$ so the angle of elevation is $$\tan^{-1} (2)\approx 63.4\degree$$ when the rocket is 1000 yd high.

#### 23.

1. $$\displaystyle d=\dfrac{50}{\tan\theta}$$
2. $$\displaystyle \theta=\tan^{-1}\left(\dfrac{50}{d} \right)$$
3. $$\theta=\tan^{-1}(0.25) \text{;}$$ the bilboard subtends an angle of $$\tan^{-1}(0.25) \approx 14\degree$$ at a distance of 200 ft.

#### 25.

1. $$\displaystyle \alpha=\tan^{-1}\left(\dfrac{1}{x}\right)$$
2. $$\displaystyle \beta=\tan^{-1}\left(\dfrac{5}{x} \right) - \tan^{-1}\left(\dfrac{1}{x}\right)$$
3. $$\beta=45\degree - \tan^{-1}\left(\dfrac{1}{5}\right) \text{,}$$ so the painting subtends an angle of $$45\degree - \tan^{-1}\left(\dfrac{1}{5}\right) \approx 33.7\degree$$ when Martin is 5 meters from the wall.

#### 27.

$$t=\dfrac{1}{2\pi\omega}\left( \sin^{-1}\dfrac{V}{V_0}-\phi \right)$$

#### 29.

$$A=\sin^{-1}\left(\dfrac{a\sin (B)}{b} \right)$$

#### 31.

$$\theta= \pm \cos^{-1}\left(\dfrac{k}{PR^4} \right)$$

#### 33.

$$\dfrac{2}{\sqrt{5}}$$

#### 35.

$$\dfrac{1}{\sqrt{5}}$$

#### 37.

$$\dfrac{5}{7}$$

#### 39.

$$\dfrac{\sqrt{1-x^2}}{x}$$

#### 41.

$$\sqrt{1-h^2}$$

#### 43.

$$\dfrac{2t}{\sqrt{4t^2+1}}$$

#### 45.

 $$x$$ $$-1$$ $$\frac{-\sqrt{3}}{2}$$ $$\frac{-\sqrt{2}}{2}$$ $$\frac{-1}{2}$$ $$0$$ $$\frac{1}{2}$$ $$\frac{\sqrt{2}}{2}$$ $$\frac{\sqrt{3}}{2}$$ $$1$$ $$\cos^{-1}(x)$$ $$\pi$$ $$\frac{5\pi}{6}$$ $$\frac{3\pi}{4}$$ $$\frac{2\pi}{3}$$ $$\frac{\pi}{2}$$ $$\frac{\pi}{3}$$ $$\frac{\pi}{4}$$ $$\frac{\pi}{6}$$ $$0$$

#### 47.

 $$x$$ $$-\sqrt{3}$$ $$-1$$ $$\frac{-1}{\sqrt{3}}$$ $$0$$ $$\frac{1}{\sqrt{3}}$$ $$1$$ $$\sqrt{3}$$ $$\cos^{-1}(x)$$ $$\frac{-\pi}{2}$$ $$\frac{-\pi}{3}$$ $$\frac{-\pi}{6}$$ $$0$$ $$\frac{\pi}{6}$$ $$\frac{\pi}{4}$$ $$\frac{\pi}{3}$$

a–b.
c. No

a.
c. No

#### 53.

$$\dfrac{8}{17}$$

#### 55.

$$\dfrac{16}{65}$$

#### 57.

$$\dfrac{4\sqrt{2} }{7}$$

#### 59.

1. $$\displaystyle \dfrac{-63}{65}$$
2. $$\displaystyle \dfrac{16}{65}$$
3. $$\displaystyle \dfrac{-33}{65}$$
4. $$\displaystyle \dfrac{56}{65}$$

#### 61.

$$1$$

#### 63.

1. $$\displaystyle \dfrac{2x}{x^2+1}$$
2. $$\displaystyle 1-x^2$$

#### 65.

$$\sin (2\theta)= \dfrac{2x\sqrt{25-x^2}}{25} \text{,}$$ $$\cos (2\theta)= \dfrac{25-2x^2}{25}$$

#### 67.

$$\arctan\left(\dfrac{x}{3}+\dfrac{3x}{2(x^2+9)}\right)$$

#### 69.

1. $$\displaystyle -1\le x\le 1$$
2. Yes.
3. All
4. $$x\lt \dfrac{-\pi}{2}$$ or $$x\gt\dfrac{\pi}{2}$$

#### 71.

1. Domain: $$-1\le x \le 1\text{,}$$ range: $$\left\{\dfrac{\pi}{2}\right\}$$
2. Let $$\theta=\sin^{-1}(x)\text{.}$$ Then $$x=\sin(\theta)= \cos\left(\dfrac{\pi}{2} - \theta \right)$$ and $$\cos^{-1}(x)= \dfrac{\pi}{2} - \theta \text{.}$$ So $$~\sin^{-1}(x)+\cos^{-1}(x) = \theta + \left(\dfrac{\pi}{2} - \theta\right) = \dfrac{\pi}{2}$$ .

#### 73.

1. $$\displaystyle \dfrac{\theta}{2}$$
2. $$\displaystyle t=\sin(\theta)$$
3. $$\displaystyle \frac{1}{2}\arcsin (t)$$

### 8.3The Reciprocal FunctionsHomework 8-3

#### 1.

$$2.203$$

#### 3.

$$0.466$$

#### 5.

$$5.883$$

#### 7.

$$1.203$$

#### 9.

$$2$$

#### 11.

$$1$$

#### 13.

$$\dfrac{-2\sqrt{3}}{3}$$

#### 15.

$$\sqrt{2}$$

#### 17.

 $$\theta$$ $$0$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{2\pi}{3}$$ $$\dfrac{3\pi}{4}$$ $$\dfrac{5\pi}{6}$$ $$\pi$$ $$\sec (\theta)$$ $$1$$ $$\dfrac{2\sqrt{3}}{3}$$ $$\sqrt{2}$$ $$2$$ undefined $$-2$$ $$-\sqrt{2}$$ $$-\dfrac{2\sqrt{3}}{3}$$ $$-1$$ $$\csc (\theta)$$ undefined $$2$$ $$\sqrt{2}$$ $$\dfrac{2\sqrt{3}}{3}$$ $$1$$ $$\dfrac{2\sqrt{3}}{3}$$ $$\sqrt{2}$$ $$2$$ undefined $$\cot (\theta)$$ undefined $$\sqrt{3}$$ $$1$$ $$\dfrac{\sqrt{3}}{3}$$ $$0$$ $$\dfrac{-\sqrt{3}}{3}$$ $$-1$$ $$-\sqrt{3}$$ undefined

#### 19.

1. $$\displaystyle 0.980$$
2. $$\displaystyle 1.020$$
3. $$\displaystyle 1.369$$
4. $$\displaystyle 1.020$$
5. $$\displaystyle 0.284$$
6. $$\displaystyle 1.020$$

#### 21.

$$\sin (\theta) = \dfrac{4}{5}\text{,}$$ $$~\cos (\theta) = \dfrac{3}{5}\text{,}$$ $$~\tan (\theta) = \dfrac{4}{3}\text{,}$$ $$~\sec (\theta) = \dfrac{5}{3}\text{,}$$ $$~\csc (\theta) = \dfrac{5}{4}\text{,}$$ $$~\cot (\theta) = \dfrac{3}{4}$$

#### 23.

$$\sin (\theta) = \dfrac{4}{\sqrt{41}}\text{,}$$ $$~\cos (\theta) = \dfrac{5}{\sqrt{41}}\text{,}$$ $$~\tan (\theta) = \dfrac{4}{5}\text{,}$$ $$~\sec (\theta) = \dfrac{\sqrt{41}}{5}\text{,}$$ $$~\csc (\theta) = \dfrac{\sqrt{41}}{4}\text{,}$$ $$~\cot (\theta) = \dfrac{5}{4}$$

#### 25.

$$\sin (\theta) = \dfrac{5}{\sqrt{74}}\text{,}$$ $$~\cos (\theta) = \dfrac{-7}{\sqrt{74}}\text{,}$$ $$~\tan (\theta) = \dfrac{-5}{7}\text{,}$$ $$~\sec (\theta) = \dfrac{-\sqrt{74}}{7}\text{,}$$ $$~\csc (\theta) = \dfrac{\sqrt{74}}{5}\text{,}$$ $$~\cot (\theta) = \dfrac{-7}{5}$$

#### 27.

$$\sin (\theta) = \dfrac{-5}{8}\text{,}$$ $$~\cos (\theta) = \dfrac{\sqrt{39}}{8}\text{,}$$ $$~\tan (\theta) = \dfrac{5}{\sqrt{39}}\text{,}$$ $$~\sec (\theta) = \dfrac{-8}{\sqrt{39}}\text{,}$$ $$~\csc (\theta) = \dfrac{-8}{5}\text{,}$$ $$~\cot (\theta) = \dfrac{\sqrt{39}}{5}$$

#### 29.

1. $$\displaystyle d=h\csc (\theta)$$
2. 155.572 miles

#### 31.

1. 0.78 sec
2. $$\displaystyle l=8t^2\sin (2\theta)$$

#### 33.

$$\sin (\theta) = \dfrac{7}{\sqrt{x^2+49}}\text{,}$$ $$~\cos (\theta) = \dfrac{x}{\sqrt{x^2+49}}\text{,}$$ $$~\tan (\theta) = \dfrac{7}{x}\text{,}$$ $$~\sec (\theta) = \dfrac{\sqrt{x^2+49}}{x}\text{,}$$ $$~\csc (\theta) = \dfrac{\sqrt{x^2+49}}{7}\text{,}$$ $$~\cot (\theta) = \dfrac{x}{7}$$

#### 35.

$$\sin (\theta) = S\text{,}$$ $$~\cos (\theta) = \sqrt{1-S^2}\text{,}$$ $$~\tan (\theta) = \dfrac{S}{\sqrt{1-S^2}}\text{,}$$ $$~\sec (\theta) = \dfrac{1}{\sqrt{1-S^2}}\text{,}$$ $$~\csc (\theta) = \dfrac{1}{S}\text{,}$$ $$~\cot (\theta) = \dfrac{\sqrt{1-S^2}}{S}$$

#### 37.

$$\sin (\theta) = \dfrac{-\sqrt{9-a^2}}{3}\text{,}$$ $$~\cos (\theta) = \dfrac{a}{3}\text{,}$$ $$~\tan (\theta) = \dfrac{-\sqrt{9-a^2}}{a}\text{,}$$ $$~\sec (\theta) = \dfrac{3}{a}\text{,}$$ $$~\csc (\theta) = \dfrac{-3}{\sqrt{9-a^2}}\text{,}$$ $$~\cot (\theta) = \dfrac{-a}{\sqrt{9-a^2}}$$

#### 39.

$$AC,~OA,~BD,~OD,~OE,~EF$$

#### 41.

$$~\sin (\theta) = \dfrac{-\sqrt{3}}{2}\text{,}$$ $$~\cos (\theta) = \dfrac{1}{2}\text{,}$$ $$~\tan (\theta) = -\sqrt{3}\text{,}$$ $$~\sec (\theta) = 2\text{,}$$ $$~\csc (\theta) = \dfrac{-2\sqrt{3}}{3}\text{,}$$ $$~\cot (\theta) = \dfrac{-\sqrt{3}}{3}$$

#### 43.

$$\sin (\alpha) = \dfrac{1}{3}\text{,}$$ $$~\cos (\alpha) = \dfrac{2\sqrt{2}}{3}\text{,}$$ $$~\tan (\alpha) = \dfrac{\sqrt{2}}{4}\text{,}$$ $$~\sec (\alpha) = \dfrac{3\sqrt{2}}{4}\text{,}$$ $$~\csc (\alpha) = 3\text{,}$$ $$~\cot (\alpha) = 2\sqrt{2}$$

#### 45.

$$\sin (\gamma) = \dfrac{-4}{\sqrt{17}}\text{,}$$ $$\cos (\gamma) = \dfrac{-1}{\sqrt{17}}\text{,}$$ $$\tan (\gamma) = 4\text{,}$$ $$\sec (\gamma) = -\sqrt{17}\text{,}$$ $$\csc (\gamma) = \dfrac{-\sqrt{17}}{4}\text{,}$$ $$\cot (\gamma) = \dfrac{1}{4}$$

#### 47.

$$\dfrac{4\sqrt{3}}{3}+2\sqrt{2}$$

#### 49.

$$\dfrac{\sqrt{3}}{3}$$

#### 51.

$$\dfrac{4\sqrt{6}}{3}+\dfrac{10}{3}$$

#### 53.

 $$x$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$\sec (x)$$ $$1$$ $$\sqrt{2}$$ undefined $$-\sqrt{2}$$ $$-1$$ $$-\sqrt{2}$$ undefined $$\sqrt{2}$$ $$1$$

#### 57.

 $$x$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$\cot (x)$$ undefined $$1$$ $$0$$ $$-1$$ undefined $$1$$ $$0$$ $$-1$$ undefined

#### 59.

\begin{aligned}[t]\dfrac{\csc (x)}{\cot (x)} \amp=\dfrac{\dfrac{1}{\sin (x)}}{\dfrac{\cos (x)}{\sin (x)}}\\ \amp= \dfrac{1}{\sin (x)}\div \dfrac{\cos (x)}{\sin (x)} \\ \amp= \dfrac{1}{\sin (x)}\cdot \dfrac{\sin (x)}{\cos (x)} \\ \amp= \dfrac{1}{\cos (x)} \\ \amp= \sec (x)\end{aligned}

#### 61.

$$\dfrac{\sec (x) \cot (x)}{\csc (x)}=\dfrac{\dfrac{1}{\cos (x)} \cdot \dfrac{\cos (x)}{\sin (x)}}{\dfrac{1}{\sin (x)}} = \dfrac{\dfrac{1}{\sin (x)}}{\dfrac{1}{\sin (x)}} = 1$$

#### 63.

$$\tan (x) \csc (x) = \dfrac{\sin (x)}{\cos (x)} \cdot \dfrac{1}{\sin (x)} = \dfrac{1}{\cos (x)} = \sec (x)$$

#### 65.

$$\dfrac{\pi}{6},~ \dfrac{5\pi}{6}$$

#### 67.

$$\dfrac{3\pi}{4},~ \dfrac{5\pi}{4}$$

#### 69.

$$\dfrac{5\pi}{6},~ \dfrac{11\pi}{6}$$

#### 71.

$$\dfrac{-\sqrt{5}}{5}$$

#### 73.

$$\dfrac{\sqrt{a^2-4}}{2}$$

#### 75.

$$\dfrac{\sqrt{w^2-1}}{-w}$$

#### 77.

$$\sec (s) = \dfrac{-5}{4}\text{,}$$ $$~\csc (s) = \dfrac{5}{3}\text{,}$$ $$~\cot (s) = \dfrac{-4}{3}$$

#### 79.

$$\sec (s) = \dfrac{1}{\sqrt{1-w^2}}\text{,}$$ $$~\csc (s) = \dfrac{1}{w}\text{,}$$ $$~\cot (s) = \dfrac{\sqrt{1-w^2}}{w}$$

#### 81.

$$\dfrac{\sin (\theta)}{\cos^2(\theta)}$$

#### 83.

$$\sec (t)$$

#### 85.

$$\dfrac{1-\sin (\beta)}{\cos (\beta)}$$

#### 87.

$$-\cos (x)$$

#### 89.

\begin{aligned}[t] \cos^2 (\theta) + \sin^2 (\theta) \amp = 1\\ \dfrac{\cos^2 (\theta)}{\cos^2 (\theta)}+\dfrac{\sin^2 (\theta)}{\cos^2 (\theta)}\amp = \dfrac{1}{\cos^2 (\theta)}\\ 1 + \tan^2 (\theta) \amp = \sec^2 (\theta) \end{aligned}

#### 91.

1. $$\displaystyle \csc (\theta) = -\sqrt{26}$$
2. $$\displaystyle \sin (\theta) = \dfrac{-\sqrt{26}}{26},~\cos (\theta) = \dfrac{-5\sqrt{26}}{26},~\tan (\theta) = \dfrac{1}{5},~\sec (\theta) = \dfrac{-\sqrt{26}}{5}$$

#### 93.

$$\cos (t) = \pm \sqrt{1-\sin^2 (t)}\text{,}$$ $$~\tan (t) = \dfrac{\pm \sin (t)}{\sqrt{1-\sin^2 (t)}}\text{,}$$ $$~\sec (t) = \dfrac{\pm 1}{\sqrt{1-\sin^2 (t)}}\text{,}$$ $$~\csc (t) = \dfrac{1}{\sin (t)}\text{,}$$ $$~ \cot (t) = \dfrac{\pm \sqrt{1-\sin^2 (t)}}{\sin (t)}$$

#### 95.

\begin{aligned}[t] \dfrac{a}{\sin (A)} \amp = \dfrac{b}{\sin (B)} = \dfrac{c}{\sin (C)}\\ a \cdot\dfrac{1}{\sin (A)} \amp = b \cdot \dfrac{1}{\sin (B)} = c \cdot \dfrac{1}{\sin (C)}\\ a \csc (A) \amp = b \csc (B) = c \csc (C) \end{aligned}

### 8.4Chapter Summary and ReviewReview Problems

False

True

False

False

#### 9.

$$\dfrac{2-\sqrt{21}}{5\sqrt{2}}$$

#### 11.

1. $$\displaystyle \dfrac{5\sqrt{33}-3}{32}$$
2. $$\displaystyle \dfrac{5\sqrt{33}-3}{\sqrt{5}(3\sqrt{3}+\sqrt{11})}$$

#### 13.

$$1$$

#### 15.

$$\dfrac{\tan (t) + \sqrt{3}}{1-\sqrt{3}\tan (t)}$$

#### 17.

1. $$\displaystyle \dfrac{4}{5}$$
2. $$\displaystyle \dfrac{3}{5}$$
3. $$\displaystyle \dfrac{4}{3}$$
4. $$\displaystyle \dfrac{24}{25}$$
5. $$\displaystyle \dfrac{-7}{25}$$
6. $$\displaystyle \dfrac{-24}{7}$$

#### 19.

$$\sin (9x)$$

#### 21.

$$\tan(2\phi - 2)$$

#### 23.

$$\sin (8\theta)$$

#### 25.

1. $$\displaystyle 1-2\sin^2(\theta) - \sin (\theta) = 1$$
2. $$\displaystyle 0,~\pi,~\dfrac{7\pi}{6},~\dfrac{11\pi}{6}$$

No

#### 29.

1. $$\displaystyle \dfrac{-\pi}{3}$$
2. $$\displaystyle \dfrac{2\pi}{3}$$

#### 31.

1. $$\displaystyle \tan^{-1}\left(\dfrac{52.8}{x}\right)$$
2. $$\displaystyle 69.25\degree,~ 27.83\degree$$

#### 33.

$$\theta = \sin^{-1}\left(\dfrac{v_y + gt}{v_0}\right)$$

#### 35.

$$\dfrac{2}{3}$$

#### 37.

$$\sqrt{1-4t^2}$$

#### 39.

Because $$\abs{\sin (\theta)} \le 1, ~\sin^{-1}(t)$$ is undefined for $$\abs{t} \gt 1\text{.}$$ If $$x \not= 0\text{,}$$ then either $$\abs{x} \gt 1$$ or $$\abs{\dfrac{1}{x}} \gt 1\text{.}$$ If $$x=0\text{,}$$ then $$\dfrac{1}{x}$$ is undefined.

#### 41.

1. $$\displaystyle 2.203$$
2. $$\displaystyle -3.236$$
3. $$\displaystyle 0.466$$

#### 43.

$$\sin (\theta) = \dfrac{13}{\sqrt{313}}\text{,}$$ $$~\cos (\theta) = \dfrac{12}{\sqrt{313}}\text{,}$$ $$~\tan (\theta) = \dfrac{13}{12}\text{,}$$ $$~\sec (\theta) = \dfrac{\sqrt{313}}{12}\text{,}$$ $$~\csc (\theta) = \dfrac{\sqrt{313}}{13}\text{,}$$ $$~\cot (\theta) = \dfrac{12}{13}$$

#### 45.

$$\sin (\theta) = \dfrac{1}{3}\text{,}$$ $$~\cos (\theta) = \dfrac{-2\sqrt{2}}{3}\text{,}$$ $$~\tan (\theta) = \dfrac{-1}{2\sqrt{2}}\text{,}$$ $$~\sec (\theta) = \dfrac{-3}{2\sqrt{2}}\text{,}$$ $$~\csc (\theta) = 3\text{,}$$ $$~\cot (\theta) = -2\sqrt{2}$$

#### 47.

$$\sin (\theta) = \dfrac{-9}{\sqrt{106}}\text{,}$$ $$~\cos (\theta) = \dfrac{-5}{\sqrt{106}}\text{,}$$ $$~\tan (\theta) = \dfrac{9}{5}\text{,}$$ $$~\sec (\theta) = \dfrac{-\sqrt{106}}{5}\text{,}$$ $$~\csc (\theta) = \dfrac{-\sqrt{106}}{9}\text{,}$$ $$~\cot (\theta) = \dfrac{5}{9}$$

#### 49.

$$\sin (\alpha) = \dfrac{-\sqrt{11}}{6}\text{,}$$ $$~\cos (\alpha) = \dfrac{-5}{6}\text{,}$$ $$~\tan (\alpha) = \dfrac{\sqrt{11}}{5}\text{,}$$ $$~\sec (\alpha) = \dfrac{-6}{5},$$$$~\csc (\alpha) = \dfrac{-6}{\sqrt{11}}\text{,}$$ $$~\cot (\alpha) = \dfrac{5}{\sqrt{11}}$$

#### 51.

$$\sin (\theta )= \dfrac{s}{4},$$ $$~\cos (\theta) = \dfrac{\sqrt{16-s^2}}{4},$$ $$~\tan (\theta) = \dfrac{s}{\sqrt{16-s^2}},$$ $$~\sec (\theta) = \dfrac{4}{\sqrt{16-s^2}},$$ $$~\csc (\theta) = \dfrac{4}{s},$$ $$~\cot (\theta) = \dfrac{\sqrt{16-s^2}}{s}$$

#### 53.

$$\sin (\theta) = \dfrac{w}{\sqrt{w^2+144}},$$ $$~\cos (\theta) = \dfrac{-12}{\sqrt{w^2+144}},$$ $$~\tan (\theta) = \dfrac{-w}{12},$$ $$~\sec (\theta) = \dfrac{-\sqrt{w^2+144}}{12},$$ $$~\csc (\theta) = \dfrac{\sqrt{w^2+144}}{w},$$ $$~\cot (\theta) = \dfrac{-12}{w}$$

#### 55.

$$\sin (\alpha) = \dfrac{k}{2},$$ $$~\cos (\alpha) = \dfrac{-\sqrt{4-k^2}}{2},$$ $$~\tan (\alpha) = \dfrac{-k}{\sqrt{4-k^2}},$$ $$~\sec (\alpha) = \dfrac{-2}{\sqrt{4-k^2}},$$ $$~\csc (\alpha) = \dfrac{2}{k},$$ $$~\cot (\alpha) = \dfrac{-\sqrt{4-k^2}}{k}$$

#### 57.

$$\sin (\theta) =0.3\text{,}$$ $$\cos (\theta) = -0.4\text{,}$$ $$\tan (\theta) = -0.75\text{,}$$ $$\sec (\theta) = -2.5\text{,}$$ $$\csc (\theta) \approx 3.33\text{,}$$ $$\cot \theta (\approx) -1.33$$

#### 59.

$$-8$$

#### 61.

$$\sqrt{2}$$

#### 63.

$$\theta \approx 2.8,~\theta \approx 0.30$$

#### 65.

$$y = \csc (x)$$ or $$y = \cot (x)$$

#### 67.

$$y = \sec (x)$$

#### 69.

$$y = \sec (x)$$ or $$y = \csc (x)$$

#### 71.

$$f(x) = \sin (x) - 1$$

#### 73.

$$G(x) = \tan (x) -1$$

#### 75.

$$\cos^2 (x)$$

#### 77.

$$\cos^2 (B)$$

#### 79.

$$\csc (\theta)$$

#### 81.

$$\sqrt{3} \tan (\theta) \sin (\theta)$$

#### 83.

1. $$\displaystyle AC = \tan (\alpha),~DC = \tan (\beta),~AD = \tan (\alpha) - \tan (\beta)$$
2. They are right triangles that share $$\angle B\text{.}$$
3. $$\angle A = \angle F,~ \angle B$$ is the complement of $$\angle A,$$ and $$\angle FDC$$ is the complement of $$\angle F\text{.}$$
4. $$\dfrac{CF}{CD} = \tan (\alpha),$$ so $$CF = \tan (\alpha) \tan (\beta)$$
5. They are right triangles with $$\angle A = \angle F\text{.}$$
6. $$\angle EBD = \alpha - \beta,$$ so $$\tan (\alpha - \beta) = \dfrac{\text{opp}}{\text{adj}} = \dfrac{DE}{BE};~~\dfrac{DE}{BE}$$ and $$\dfrac{AD}{BF}$$ are ratios of corresponding sides of similar triangles; $$AD = \tan (\alpha) - \tan (\beta)$$ by part (a), $$BF = BC + CF = 1 + \tan (\alpha) \tan (\beta)$$ by part (d).

#### 85.

$$d=25\csc (112\degree),~\alpha = 45\degree,~a \approx 19.07,~b \approx 10.54$$

### 9Vectors9.1Geometric FormHomework 9-1

#### 7.

$$\bf{A}$$ and $$\bf{E}$$

#### 9.

$$\bf{H}$$ and $$\bf{K}$$

#### 19.

$$\|{\bf{A}}\| = \sqrt{13},~ \theta = -33.7\degree$$

#### 21.

$$\|{\bf{C}}\| = 1,~ \theta = 90\degree$$

#### 23.

$$\|{\bf{E}}\| = 5,~ \theta = 90\degree$$

#### 25.

$$\|{\bf{G}}\| = 4,~ \theta = 180\degree$$

#### 27.

$$\|{\bf{v}}\| = 13,~ \theta = -67.38\degree$$

#### 29.

$$\|{\bf{v}}\| = \sqrt{85} \approx 9.22,~ \theta = 229.4\degree$$

#### 31.

$$\|{\bf{v} + \bf{w}}\| = 32.9,~ \theta = 109.3\degree$$

#### 33.

$$\|{\bf{v} + \bf{w}}\| = 11.4,~ \theta = 162.4\degree$$

#### 35.

4.47 mi, $$23.4\degree$$ east of north

#### 37.

129.4 mph, $$85.4\degree$$ west of north

#### 39.

1. $$\displaystyle v_x = 10,~ v_y = 10\sqrt{3},~ w_x = 5\sqrt{2},~ w_y = -5\sqrt{2}$$
2. 19.9 mph, $$59\degree$$ east of north

#### 41.

1. $$\displaystyle v_x \approx -1.23,~ v_y \approx 3.38,~ w_x \approx -0.32,~ w_y \approx -2.23$$
2. 1.9 km, $$54.5\degree$$ west of north

#### 51.

$$u_x = 2\text{,}$$ $$~ u_y = 1\text{,}$$ $$~ v_x = 1\text{,}$$$$~ v_y = -3\text{,}$$ $$~ A_x = 1\text{,}$$ $$~ A_y = 4\text{;}$$ $$~ A_x = u_x - v_x\text{,}$$ $$~ A_y = u_y - v_y$$

### 9.2Coordinate FormHomework 9-2

#### 1.

$${\bf{u}} = 3{\bf{i}}+2{\bf{j}}$$
1. $$\displaystyle \sqrt{13}$$
2. $$\displaystyle 6{\bf{i}}+4{\bf{j}}$$
3. $$\displaystyle 2\sqrt{13}$$

#### 3.

$${\bf{w}} = 6{\bf{i}}-3{\bf{j}}$$
1. $$\displaystyle 3\sqrt{5}$$
2. $$\displaystyle -6{\bf{i}}+3{\bf{j}}$$
3. $$\displaystyle 3\sqrt{5}$$

#### 5.

1. $${\bf{u}}+{\bf{v}} = -2{\bf{i}}+5{\bf{j}}$$ and $$\|{\bf{u}}+{\bf{v}}\| = \sqrt{29}$$
2. $$\displaystyle \|{\bf{u}}\|+\|{\bf{v}}\| \ge \|{\bf{u}}+{\bf{v}}\|$$

#### 7.

1. $$-5{\bf{i}}+8{\bf{j}}$$
2. $$\displaystyle \|{\bf{v}}\| = \sqrt{89},~~\theta = 122\degree$$

#### 9.

1. $$-2{\bf{i}}-{\bf{j}}$$
2. $$\displaystyle \|{\bf{v}}\| = \sqrt{5},~~\theta = 206.6\degree$$

#### 11.

1. $$\displaystyle 18{\bf{i}}+12{\bf{j}}$$
2. $$\displaystyle \|{\bf{v}}\| = 6\sqrt{13},~~\theta = 33.7\degree$$

#### 13.

$$\|{\bf{v}}\| = 6\sqrt{2},~~\theta = 135\degree$$

#### 15.

$$\|{\bf{w}}\| = 14,~~\theta = -30\degree$$

#### 17.

$$\|{\bf{q}}\| = 4\sqrt{745},~~\theta = 61.56\degree$$

#### 19.

$${\bf{v}} = 3\sqrt{2}{\bf{i}}-3\sqrt{2}{\bf{j}}$$

#### 21.

$${\bf{v}} \approx 6.629{\bf{i}}+4.995{\bf{j}}$$

#### 23.

$${\bf{i}}-2{\bf{j}}$$

#### 25.

$$-4{\bf{i}}+4{\bf{j}}$$

#### 27.

$$12{\bf{i}}+3{\bf{j}}$$

#### 29.

$$2.8{\bf{i}}+1.9{\bf{j}}$$

#### 31.

$$-3{\bf{i}}+7{\bf{j}}$$

#### 33.

$$-8{\bf{i}}-20{\bf{j}}$$

#### 35.

$$14{\bf{i}}-9{\bf{j}}$$

#### 37.

$$-9{\bf{i}}+23{\bf{j}}$$

#### 39.

$$\dfrac{-12}{13}{\bf{i}}+\dfrac{5}{13}{\bf{j}}$$

#### 41.

$$\dfrac{1}{\sqrt{2}}{\bf{i}}-\dfrac{1}{\sqrt{2}}{\bf{j}}$$

#### 43.

$$24{\bf{i}}+45{\bf{j}}$$

#### 45.

$$\dfrac{-12}{\sqrt{10}}{\bf{i}}+\dfrac{4}{\sqrt{10}}{\bf{j}}$$

#### 47.

1. $$\displaystyle {\bf{u}}=2.393{\bf{i}}+1.016{\bf{j}},~~{\bf{v}}=-4.242{\bf{i}}-3.956{\bf{j}}$$
2. $$\displaystyle -1.849{\bf{i}}-2.940{\bf{j}}$$

#### 49.

1. $$\displaystyle {\bf{u}}=-11.97{\bf{i}}+32.889{\bf{j}},~~{\bf{v}}=-57.955{\bf{i}}+15.529{\bf{j}}$$
2. $$\displaystyle 45.98{\bf{i}}+17.36{\bf{j}}$$

#### 51.

1. 1700 m, $$28.1\degree$$ east of south

#### 53.

1. 21.98 km, $$2.27\degree$$ north of west

#### 55.

1. 83 mi, $$62\degree$$ east of north

#### 57.

1. $$\displaystyle -4{\bf{i}}-5{\bf{j}}$$
2. $$\displaystyle 4{\bf{i}}+5{\bf{j}}$$

#### 59.

1. $$\displaystyle {\bf{i}}-3{\bf{j}}$$
2. $$\displaystyle -{\bf{i}}+3{\bf{j}}$$

#### 61.

1. $$\displaystyle \|{\bf{v}}\| = 10,~ 2\|{\bf{v}}\| = 20 = 2 \cdot 10$$
2. $$\displaystyle \|k{\bf{v}}\| = \sqrt{(ka)^2 +(kb)^2} = k\sqrt{a^2 + b^2}$$

### 9.3The Dot ProductHomework 9-3

#### 1.

$$\dfrac{33}{\sqrt{13}}$$

#### 3.

$$\dfrac{-1}{\sqrt{2}}$$

#### 5.

$$-2\sqrt{5}$$

#### 7.

1. $$\displaystyle {\bf{w}} = \left(\dfrac{56}{13}{\bf{i}}+\dfrac{84}{13}{\bf{j}}\right) + \left(\dfrac{48}{13}{\bf{i}}-\dfrac{32}{13}{\bf{j}}\right)$$

#### 9.

1. $$\displaystyle {\bf{w}} = (4{\bf{i}}-4{\bf{j}}) + (2{\bf{i}}+2{\bf{j}})$$

#### 11.

$$22$$

#### 13.

$$0$$

#### 15.

$$12$$

#### 17.

$$-318.2$$

not orthogonal

orthogonal

#### 23.

$$4.4\degree$$

#### 25.

$$97.1\degree$$

#### 27.

$$8$$

#### 29.

$$-10$$

#### 31.

$$-21$$

#### 33.

$$42{\bf{i}}-28{\bf{j}}$$

#### 35.

$$4$$

38.57 lbs

1289 lbs

#### 41.

1. $$\displaystyle \dfrac{1}{\sqrt{2}}{\bf{i}}+\dfrac{1}{\sqrt{2}}{\bf{j}}~~\text{and}~~\dfrac{-1}{\sqrt{2}}{\bf{i}}+\dfrac{1}{\sqrt{2}}{\bf{j}}$$
2. $$\displaystyle {\bf{u}} \cdot {\bf{v}} = 0$$
3. $$\dfrac{11}{\sqrt{2}}$$ and $$\dfrac{5}{\sqrt{2}}$$

#### 43.

$${\bf{v}} \cdot {\bf{v}} = c^2 + d^2$$

#### 45.

$$k{\bf{u}} \cdot {\bf{v}} = kac+kbd = k(ac+bd) = (akc + bkd)$$

#### 47.

\begin{aligned}[t] ({\bf{u}}-{\bf{v}}) \cdot ({\bf{u}}+{\bf{v}}) \amp= (a-c)(a+c)+(b-d)(b+d)\\ \amp= (a^2+b^2)-(c^2+d^2)\end{aligned}

#### 49.

$$\dfrac{a \cdot 1 + b \cdot 0}{1} = a$$ and $$\dfrac{a \cdot 0 + b \cdot 1}{1} = b$$

#### 51.

1. Both $${\bf{i}} \cdot {\bf{i}}=1$$ and $${\bf{j}} \cdot {\bf{j}}=1$$ because $$1 \cdot 1 \cos 0 = 1\text{;}$$ $${\bf{i}} \cdot {\bf{j}} = 1 \cdot 1 \cos 90\degree =0$$
2. $$\displaystyle (a{\bf{i}}+b{\bf{j}}) \cdot (c{\bf{i}}+d{\bf{j}}) = ac(1) + ad(0) + bc(0) + bd(1) = ac+bd$$

#### 53.

1. $$\displaystyle \|{\bf{u}}-{\bf{v}}\|^2 = {\bf{u}} \cdot {\bf{u}} - 2{\bf{u}} \cdot {\bf{v}} +{\bf{v}} \cdot {\bf{v}}= \|{\bf{u}}\|^2+\|{\bf{v}}\|^2 - 2\|{\bf{u}}\|\|{\bf{v}}\|\cos \theta$$
2. Let $$a = \|{\bf{u}}\|,~ b = \|{\bf{v}}\|,~c = \|{\bf{u}}-{\bf{v}}\|\text{,}$$ and $$C = \theta$$

### 9.4Chapter Summary and ReviewReview Problems

#### 1.

$$v_N=8.45$$ mph, $$v_E=-18.13$$ mph

#### 3.

$$v_N=-1127.63$$ lbs, $$v_E=-410.42$$ lbs

#### 5.

$$\|{\bf{A}}\|=10.9,~\theta = 236.3\degree$$

#### 7.

$${\bf{i}}-\sqrt{3}{\bf{j}}$$

#### 9.

1. $$15{\bf{i}}+3{\bf{j}}$$
2. $$\displaystyle \|{\bf{v}}\|=15.3,~\theta = 11.3\degree$$

#### 11.

1. $$2{\bf{i}}-6{\bf{j}}$$
2. $$\displaystyle \|{\bf{v}}\|=6.3~\text{mi},~\theta = 288.4\degree$$

#### 13.

1. $$\displaystyle 7.64~\text{km},~\theta = 30.31\degree$$

#### 15.

1. $$\displaystyle 8.46~\text{mi},~\theta = 155.6\degree$$

#### 17.

1. $${\bf{F_1}}=-200{\bf{i}},~{\bf{F_2}}=-60\sqrt{2}{\bf{i}}-60\sqrt{2}{\bf{j}},~{\bf{F_3}}=50\sqrt{3}{\bf{i}}+50{\bf{j}},$$ $${\bf{F_4}}=-125{\bf{i}}+125\sqrt{3}{\bf{j}}$$
2. $$\displaystyle -73.25{\bf{i}}+181.65{\bf{j}}$$

#### 19.

$$13{\bf{i}}+5{\bf{j}}$$

#### 21.

$$-7{\bf{i}}-14{\bf{j}}$$

#### 23.

$$\dfrac{2}{\sqrt{13}}{\bf{i}}+\dfrac{3}{\sqrt{13}}{\bf{j}}$$

#### 25.

$$\dfrac{-6}{\sqrt{29}}{\bf{i}}-\dfrac{15}{\sqrt{29}}{\bf{j}}$$

#### 27.

$$-3.45$$

#### 29.

$$-8.08$$

#### 31.

$$106.26\degree$$

### 10Polar Coordinates and Complex Numbers10.1Polar CoordinatesHomework 10-1

#### 9.

$$\left(5, \dfrac{3\pi}{4}\right)$$

#### 11.

$$(1, \pi)$$

#### 13.

$$\left(3, \dfrac{4\pi}{3}\right)$$

#### 15.

$$\left(2, \dfrac{\pi}{12}\right)$$

#### 17.

$$(-3, 3\sqrt{3})$$

#### 19.

$$\left(\dfrac{3}{\sqrt{2}}, \dfrac{-3}{\sqrt{2}}\right)$$

#### 21.

$$(-2.15, -1.06)$$

#### 23.

$$(-0.14, -1.99)$$

#### 25.

$$\left(7\sqrt{2}, \dfrac{\pi}{4}\right)$$

#### 27.

$$\left(2\sqrt{2}, \dfrac{11\pi}{6}\right)$$

#### 29.

$$\left(\sqrt{13}, \pi+\tan^{-1}\dfrac{2}{3}\right)$$

#### 31.

$$(2, \pi)$$

#### 33.

1. $$\displaystyle \left(-2, \dfrac{11\pi}{6}\right)$$
2. $$\displaystyle \left(2, \dfrac{-7\pi}{6}\right)$$

#### 35.

1. $$\displaystyle (-3,0)$$
2. $$\displaystyle (3, -\pi)$$

#### 37.

1. $$\displaystyle (-2.3, 2.06)$$
2. $$\displaystyle (2.3, -1.08)$$

#### 45.

$$r \ge 0,~ \dfrac{\pi}{6} \le \theta \le \dfrac{\pi}{3}$$

#### 47.

$$r \ge 1,~ \dfrac{\pi}{2} \le \theta \le \pi$$

#### 49.

$$-1 \le r \le 1,~ \dfrac{3\pi}{4} \le \theta \le \pi$$

#### 51.

$$x^2+y^2=2$$

#### 53.

$$x^2+y^2=4x$$

#### 55.

$$y=1$$

#### 57.

$$y=2x$$

#### 59.

$$x^2+y^2=3x$$

#### 61.

$$x^2=4-4y$$

#### 63.

$$2x+y=1$$

#### 65.

$$r=2\sec (\theta)$$

#### 67.

$$2r^2=\sec (\theta) \csc (\theta)$$

#### 69.

$$r=4\cot (\theta) \csc (\theta)$$

#### 71.

$$r=4$$

#### 73.

\begin{align*} d \amp =\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\ \amp = \sqrt{(r_2\cos(\theta_2)-r_1\cos(\theta_1))^2 +(r_2\sin(\theta_2)-r_1\sin(\theta_1))^2}\\ \amp = \sqrt{r_2^2\cos^2(\theta_2) - 2r_2r_1\cos(\theta_2)\cos(\theta_1)+r_1^2\cos^2(\theta_1)+r_2^2\sin^2(\theta_2) - 2r_2r_1\sin(\theta_2)\sin(\theta_1)+r_1^2\sin^2(\theta_1)}\\ \amp = \sqrt{r_2^2+r_1^2 - 2r_2r_1(\cos(\theta_2)\cos(\theta_1)-\sin(\theta_2)\sin(\theta_1))}\\ \amp = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos (\theta_2 - \theta_1)} \end{align*}

### 10.2Polar GraphsHomework 10-2

#### 1.

1. $$k$$ is the radius
2. $$\displaystyle x^2+y^2=1,~ x^2+y^2=4,~ x^2+y^2=9$$

#### 3.

1. $$\tan k$$ is the slope
2. $$\displaystyle y=\dfrac{x}{\sqrt{3}},~ y=\sqrt{3}x,~ y=-\sqrt{3}x,~ y=\dfrac{x}{\sqrt{3}}$$

#### 5.

 $$\theta$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$r=2$$ $$2$$ $$2$$ $$2$$ $$2$$ $$2$$ $$2$$ $$2$$ $$2$$
 $$\theta$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$r=-2$$ $$-2$$ $$-2$$ $$-2$$ $$-2$$ $$-2$$ $$-2$$ $$-2$$ $$-2$$
The graph of $$r=2$$ begins at the right-most point (and proceeds counter-clockwise); the graph of $$r=-2$$ begins at the left-most point.

#### 7.

1.  $$\theta$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$r$$ $$-4$$ $$-2\sqrt{2}$$ $$0$$ $$2\sqrt{2}$$ $$4$$
The graph is traced again.
2. center: $$(2,0)\text{,}$$ radius: $$2$$
3. $$\displaystyle (x-2)^2+y^2=4$$

#### 9.

1. For $$a \gt 0\text{,}$$ $$a$$ is the radius of a circle centerd on the positive $$y$$-axis; for $$a \lt 0\text{,}$$ $$\abs{a}$$ is the radius of a circle centerd on the negative $$y$$-axis.

#### 11.

1.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$1$$ $$2$$ $$1$$ $$0$$ $$1$$
2.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$-1$$ $$0$$ $$-1$$ $$-2$$ $$-1$$
3.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$1$$ $$0$$ $$1$$ $$2$$ $$1$$
4.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$-1$$ $$-2$$ $$-1$$ $$0$$ $$-1$$

#### 13.

1.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$3$$ $$2$$ $$1$$ $$2$$ $$3$$
2.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$1$$ $$2$$ $$3$$ $$2$$ $$1$$
3.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$3$$ $$1$$ $$-1$$ $$1$$ $$3$$
4.  $$\theta$$ $$0$$ $$\dfrac{\pi}{2}$$ $$\pi$$ $$\dfrac{3\pi}{2}$$ $$2\pi$$ $$r$$ $$-1$$ $$1$$ $$3$$ $$1$$ $$-1$$

#### 15.

1. There are $$n$$ petals if $$n$$ is odd, and $$2n$$ petals if $$n$$ is even.
2. $$n=2:~ \dfrac{\pi}{4},$$ $$~ \dfrac{3\pi}{4},~$$$$\dfrac{5\pi}{4},~$$$$\dfrac{7\pi}{4};~$$ $$n=3:~\dfrac{\pi}{6},~$$$$\dfrac{5\pi}{6},~$$$$\dfrac{3\pi}{2};$$ $$n=4:~ \dfrac{\pi}{8},~$$$$\dfrac{3\pi}{8},~$$$$\dfrac{5\pi}{8},~$$$$\dfrac{7\pi}{8},~$$$$\dfrac{9\pi}{8},~$$$$\dfrac{11\pi}{8},~$$$$\dfrac{13\pi}{8},~$$$$\dfrac{15\pi}{8};$$ $$n=5:~~\dfrac{\pi}{10},~$$$$\dfrac{\pi}{2},~$$$$\dfrac{9\pi}{10},~$$$$\dfrac{13\pi}{10},~$$$$\dfrac{17\pi}{10}$$
3. $$a$$ is the length of the petal.

#### 17.

1. $$\displaystyle r=\pm 3\sqrt{\cos 2\theta}$$
2. $$a$$ is the length of the loop.

#### 21.

1.  $$\theta$$ $$0$$ $$\dfrac{\pi}{12}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{5\pi}{12}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{7\pi}{12}$$ $$\dfrac{2\pi}{3}$$ $$3\theta$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$r$$ $$0$$ $$\dfrac{\sqrt{2}}{2}$$ $$1$$ $$\dfrac{\sqrt{2}}{2}$$ $$0$$ $$\dfrac{-\sqrt{2}}{2}$$ $$-1$$ $$\dfrac{-\sqrt{2}}{2}$$ $$0$$
2.  $$\theta$$ $$0$$ $$\dfrac{\pi}{12}$$ $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{5\pi}{12}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{7\pi}{12}$$ $$\dfrac{2\pi}{3}$$ $$3\theta$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$r$$ $$0$$ $$\dfrac{\sqrt{2}}{2}$$ $$1$$ $$\dfrac{\sqrt{2}}{2}$$ $$0$$ $$\dfrac{-\sqrt{2}}{2}$$ $$-1$$ $$\dfrac{-\sqrt{2}}{2}$$ $$0$$

#### 23.

1.  $$\theta$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$y$$ $$4$$ $$2+\sqrt{2}$$ $$2$$ $$2-\sqrt{2}$$ $$0$$ $$2-\sqrt{2}$$ $$2$$ $$2+\sqrt{2}$$ $$4$$
2.  $$\theta$$ $$0$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{2}$$ $$\dfrac{3\pi}{4}$$ $$\pi$$ $$\dfrac{5\pi}{4}$$ $$\dfrac{3\pi}{2}$$ $$\dfrac{7\pi}{4}$$ $$2\pi$$ $$y$$ $$4$$ $$2+\sqrt{2}$$ $$2$$ $$2-\sqrt{2}$$ $$0$$ $$2-\sqrt{2}$$ $$2$$ $$2+\sqrt{2}$$ $$4$$

circle

line

circle

cardioid

limaçon

rose

rose

limaçon

lemniscate

circle

arcs of a circle

semicircle

rose

cardioid

parabola

ellipse

hyperbola

#### 59.

$$r=2+2\cos (\theta)$$

#### 61.

$$r=3\sin (5\theta)$$

#### 63.

$$r=5\sin (\theta)$$

#### 65.

$$r=1+2\cos (\theta)$$

#### 67.

$$(0,0)\text{,}$$$$~ \left(\dfrac{1}{2},\dfrac{\pi}{3}\right)\text{,}$$$$~ \left(\dfrac{1}{2},\dfrac{5\pi}{3}\right)$$

#### 69.

$$(0,0)\text{,}$$ $$~ \left(\dfrac{3}{\sqrt{2}}, \dfrac{\pi}{4}\right)\text{,}$$ $$~ \left(\dfrac{-3}{\sqrt{2}}, \dfrac{5\pi}{4}\right)$$

#### 71.

$$\left(1, \dfrac{\pi}{2}\right)\text{,}$$ $$~\left(1, \dfrac{3\pi}{2}\right)$$

#### 73.

$$\left(\dfrac{4+\sqrt{2}}{2},\dfrac{3\pi}{4}\right)\text{,}$$ $$~\left(\dfrac{4-\sqrt{2}}{2},\dfrac{7\pi}{4}\right)$$

#### 83.

The curve has $$n$$ large loops and $$n$$ small loops.

### 10.3Complex NumbersHomework 10-3

#### 1.

1. $$\displaystyle 5i-4$$
2. $$\displaystyle -4+i$$
3. $$\displaystyle \dfrac{-5}{6}-\dfrac{\sqrt{2}}{6}i$$

#### 3.

$$-3\pm 2i$$

#### 5.

$$\dfrac{1}{6} \pm \dfrac{\sqrt{11}}{6}i$$

#### 7.

$$13+4i$$

#### 9.

$$-0.8+3.8i$$

#### 11.

$$20-10i$$

#### 13.

$$-14+34i$$

#### 15.

$$46+14i\sqrt{3}$$

#### 17.

$$52$$

#### 19.

$$-2-2i$$

#### 21.

$$-1+4i$$

#### 23.

$$\dfrac{35}{3}+\dfrac{20}{3}i$$

#### 25.

$$\dfrac{-25}{29}+\dfrac{10}{29}i$$

#### 27.

$$\dfrac{3}{4}-\dfrac{\sqrt{3}}{4}i$$

#### 29.

$$\dfrac{-2}{3}+\dfrac{\sqrt{5}}{3}i$$

#### 31.

$$i$$

#### 33.

1. $$\displaystyle -1$$
2. $$\displaystyle 1$$
3. $$\displaystyle -i$$
4. $$\displaystyle -1$$

#### 35.

1. $$\displaystyle 0$$
2. $$\displaystyle 0$$

#### 37.

1. $$\displaystyle 0$$
2. $$\displaystyle 0$$

#### 39.

1. $$\displaystyle 0$$
2. $$\displaystyle 0$$

#### 41.

$$4z^2+49$$

#### 43.

$$x^2+6x+10$$

#### 45.

$$v^2-8v+17$$

#### 59.

$$(a+bi)(c+di) = ac+adi+bci+bdi^2=(ac-bd)+(ad+bc)i$$

#### 61.

\begin{align*} z_1+z_2 \amp =(a+bi)+(c+di)=(a+c)+(b+d)i\\ \amp =(c+a)+(d+b)i=(c+di)+(a+bi)=z_2+z_1 \end{align*}
\begin{align*} z_1z_2 \amp =(a+bi)(c+di) = (ac-bd)+(ad+bc)i\\ \amp =(ca-db)+(da+cb)i=z_2z_1 \end{align*}

#### 63.

1. $$z+\bar{z}=(a+bi)+(a-bi)=2a;~~~$$ $$z-\bar{z}=(a+bi)-(a-bi)=-2bi$$
2. $$\displaystyle z\bar{z}=(a+bi)(a-bi)=a^2+b^2=\abs{z}^2$$

#### 65.

No. Let $$t=i$$ and $$z=-i\text{.}$$ Then $$w=t+z=i-i=0,$$ so $$\abs{w}=0,$$ but $$\abs{t}+\abs{z}=\abs{i}+\abs{-i}=1+1=2.$$

#### 67.

1. $$\displaystyle 2-\sqrt{5}$$
2. $$\displaystyle x^2-4x-1=0$$

#### 69.

1. $$\displaystyle 4+3i$$
2. $$\displaystyle x^2-8x+25=0$$

#### 71.

$$x^4-6x^3+23x^2-50x+50=0$$

#### 73.

$$x^4-7x^3+20x^2-19x+13=0$$

### 10.4Polar Form for Complex NumbersHomework 10-4

#### 1.

$$1\text{,}$$ $$~i\text{,}$$ $$~-1\text{,}$$$$~-i\text{,}$$ $$~1$$

#### 3.

$$1+2i\text{,}$$ $$-2+i$$

#### 5.

$$-3+3i\sqrt{3}$$

#### 7.

$$-1+i$$

#### 9.

$$2.34-4.21i$$

#### 11.

$$-5.07+10.88i$$

#### 13.

$$3\left(\cos \left(\dfrac{\pi}{2}\right) + i\sin \left(\dfrac{\pi}{2}\right)\right)\text{,}$$ $$3\left(\cos \left(\dfrac{3\pi}{2}\right) + i\sin \left(\dfrac{3\pi}{2}\right)\right)$$

#### 15.

$$2\sqrt{3}\left(\cos \left(\dfrac{7\pi}{6}\right) + i\sin \left(\dfrac{7\pi}{6}\right)\right)\text{,}$$ $$2\sqrt{3}\left(\cos \left(\dfrac{11\pi}{6}\right) + i\sin \left(\dfrac{11\pi}{6}\right)\right)$$

#### 17.

$$4.47(\cos (2.68) + i\sin (2.68)),~$$ $$4.47(\cos (5.82) + i\sin (5.82))$$

#### 19.

$$8.60(\cos (5.78) + i\sin (5.78)),~$$ $$8.60(\cos (0.51) + i\sin (0.51))$$

#### 21.

$$5(\cos (0.93) + i\sin (0.93)),~$$ $$5(\cos (5.36) + i\sin (5.36)),~$$ $$5(\cos (2.21) + i\sin (2.21)),~$$ $$5(\cos (4.07) + i\sin (4.07))$$

#### 23.

If $$z=r(\cos (\theta)+ i\sin (\theta)),$$ then $$\bar{z}=r(\cos (2\pi -\theta) + i\sin (2\pi -\theta)$$

#### 25.

$$z_1z_2 = 2\left(\cos \left(\dfrac{\pi}{6}\right) + i\sin \left(\dfrac{\pi}{6}\right)\right) = \sqrt{3} + i\text{;}$$$$~~\dfrac{z_1}{z_2}=8\left(\cos \left(\dfrac{\pi}{2}\right) + i\sin \left(\dfrac{\pi}{2}\right)\right) = 8i$$

#### 27.

$$z_1z_2 = 6\left(\cos \left(\dfrac{9\pi}{10}\right) + i\sin \left(\dfrac{9\pi}{10}\right)\right)\text{;}$$ $$~~\dfrac{z_1}{z_2}=\dfrac{3}{2}\left(\cos \left(\dfrac{3\pi}{10}\right) + i\sin \left(\dfrac{3\pi}{10}\right)\right)$$

#### 29.

$$z_1z_2 = 8\text{;}$$ $$~\dfrac{z_1}{z_2}=\dfrac{1}{2}$$

#### 31.

$$z_1z_2 = 4\sqrt{2}(\cos \dfrac{7\pi}{12} + i\sin \dfrac{7\pi}{12})\text{;}$$ $$~~\dfrac{z_1}{z_2}=2\sqrt{2}(\cos \dfrac{13\pi}{12} + i\sin \dfrac{13\pi}{12})$$

#### 33.

$$-128-128i$$

#### 35.

$$-128-128\sqrt{3}i$$

#### 37.

$$512+512\sqrt{3}i$$

#### 39.

$$\dfrac{1}{4}+\dfrac{1}{4}i$$

#### 41.

$$\dfrac{-\sqrt{2}}{8}-\dfrac{\sqrt{6}}{8}i$$

#### 43.

1. $$3(\cos \dfrac{\pi}{4} + i\sin \dfrac{\pi}{4})\text{,}$$ $$~3(\cos \dfrac{3\pi}{4} + i\sin \dfrac{3\pi}{4})$$
2. $$\dfrac{3}{\sqrt{2}}+\dfrac{3}{\sqrt{2}}i\text{,}$$ $$~\dfrac{-3}{\sqrt{2}}-\dfrac{3}{\sqrt{2}}i$$

#### 45.

1. $$2,~2\left(\cos \dfrac{2\pi}{5} + i\sin \dfrac{2\pi}{5}\right)\text{,}$$ $$~2\left(\cos \dfrac{4\pi}{5} + i\sin \dfrac{4\pi}{5}\right)$$ , $$~2\left(\cos \dfrac{6\pi}{5} + i\sin \dfrac{6\pi}{5}\right)\text{,}$$ $$2\left(\cos \dfrac{8\pi}{5} + i\sin \dfrac{8\pi}{5}\right)$$
2. $$2,~0.618+1.9i\text{,}$$ $$-1.618+1.176i\text{,}$$ $$-1.618-1.176i\text{,}$$ $$0.618-1.902i$$

#### 47.

1. $$4\left(\cos \dfrac{\pi}{18} + i\sin \dfrac{\pi}{18}\right)\text{,}$$ $$~4\left(\cos \dfrac{13\pi}{18} + i\sin \dfrac{13\pi}{18}\right)\text{,}$$ $$~4\left(\cos \dfrac{25\pi}{18} + i\sin \dfrac{25\pi}{18}\right)$$
2. $$1.97+0.347i\text{,}$$ $$~-1.286+1.532i\text{,}$$ $$~-0.684-1.879i$$

#### 49.

$$\abs{z} = \abs{\cos (\theta) + i\sin (\theta)} = \sqrt{\cos^2(\theta) + \sin^2(\theta)} = 1$$

#### 51.

1. $$\displaystyle 1,~(\cos \dfrac{2\pi}{3} + i\sin \dfrac{2\pi}{3}),~(\cos \dfrac{4\pi}{3} + i\sin \dfrac{4\pi}{3})$$
2. $$\displaystyle 1,~i,~-1,~-i$$
3. $$1,~(\cos \dfrac{2\pi}{5} + i\sin \dfrac{2\pi}{5}),~(\cos \dfrac{4\pi}{5} + i\sin \dfrac{4\pi}{5}),$$ $$(\cos \dfrac{6\pi}{5} + i\sin \dfrac{6\pi}{5}),~(\cos \dfrac{8\pi}{5} + i\sin \dfrac{8\pi}{5})$$
4. $$1,~(\cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}),~(\cos \dfrac{2\pi}{3} + i\sin \dfrac{2\pi}{3}),~-1,$$ $$(\cos \dfrac{4\pi}{3} + i\sin \dfrac{4\pi}{3}),~(\cos \dfrac{5\pi}{3} + i\sin \dfrac{5\pi}{3})$$

#### 53.

$$(\omega_k)^n = 1^n \left(\cos \left(n \cdot \dfrac{2\pi k}{n}\right) + i\sin \left(n \cdot \dfrac{2\pi k}{n}\right)\right) = 1(\cos 2\pi k + i\sin 2\pi k) = 1$$

#### 55.

$$8^{1/4}\left(\cos \left(\dfrac{3\pi}{8}\right) + i\sin \left(\dfrac{3\pi}{8}\right)\right)\text{,}$$ $$~8^{1/4}\left(\cos \left(\dfrac{5\pi}{8}\right) + i\sin \left(\dfrac{5\pi}{8}\right)\right)\text{,}$$ $$~8^{1/4}\left(\cos \left(\dfrac{11\pi}{8}\right) + i\sin \left(\dfrac{11\pi}{8}\right)\right)\text{,}$$ $$~8^{1/4}\left(\cos \left(\dfrac{13\pi}{8}\right) + i\sin \left(\dfrac{13\pi}{8}\right)\right)$$

#### 57.

$$\sqrt{2}\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{\pi}{3}\right) + i\sin \left(\dfrac{\pi}{3}\right)\right)\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{2\pi}{3}\right) + i\sin \left(\dfrac{2\pi}{3}\right)\right)\text{,}$$ $$~-\sqrt{2}\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{4\pi}{3}\right) + i\sin \left(\dfrac{4\pi}{3}\right)\right)\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{5\pi}{3}\right) + i\sin \left(\dfrac{5\pi}{3}\right)\right)$$

#### 59.

$$\sqrt{2}\left(\cos \left(\dfrac{\pi}{3}\right) + i\sin \left(\dfrac{\pi}{3}\right)\right)\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{2\pi}{3}\right) + i\sin \left(\dfrac{2\pi}{3}\right)\right)\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{4\pi}{3}\right) + i\sin \left(\dfrac{4\pi}{3}\right)\right)\text{,}$$ $$~\sqrt{2}\left(\cos \left(\dfrac{5\pi}{3}\right) + i\sin \left(\dfrac{5\pi}{3}\right)\right)$$
1. $$\displaystyle \cos^2 (\theta) - \sin^2 (\theta) +(2\sin (\theta) \cos (\theta))i$$
2. $$\displaystyle \cos (2\theta) + i\sin (2\theta)$$
3. $$\displaystyle \sin (2\theta) = 2\sin (\theta) \cos (\theta);~~\cos (2\theta) = \cos^2 (\theta) - \sin^2 (\theta)$$
1. $$\displaystyle \dfrac{b}{a}$$