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Section 5.1 Algebra with Trigonometric Ratios

In this chapter we apply some techniques from algebra to analyze more complicated trigonometric expressions. Before we begin, let's review some algebraic terminology.

  • An algebraic expression is any meaningful collection of numbers, variables, and operation symbols. For example, the height of a golf ball is given in feet by the expression \(-16t^2 + 64t\text{,}\) where \(t\) is the number of seconds after the ball is hit.
  • We evaluate an expression by substituting a specific value for the variable or variables involved. Thus, after 1 second, the height of the golf ball is
    \begin{equation*} -16(\alert{1})^2 + 64(1\alert{1}) = -16 + 64 = 48~ \text{feet} \end{equation*}
    and after 2 seconds, the height is
    \begin{equation*} -16(\alert{2})^2 + 64(\alert{2}) = -64 + 128 = 64~ \text{feet} \end{equation*}
    and so on.

Subsection Evaluating Trigonometric Expressions

Trigonometric ratios represent numbers, and they may appear as part of an algebraic expression. Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions.

Example 5.1

Evaluate each expression for \(X = 30\degree\) and \(Y = 135\degree\text{.}\)

  1. \(2 \tan Y + 3\sin X\)
  2. \(6\tan X \cos Y\)
Solution
  1. Substituting the values for \(X\) and \(Y\text{,}\) we get
    \begin{equation*} 2 \tan \alert{135\degree} + 3\sin \alert{30\degree} \end{equation*}
    Next, we evaluate each trig ratio and follow the order of operations.
    \begin{equation*} \begin{aligned}[t] 2 \tan \alert{135\degree} + 3\sin \alert{30\degree} \amp = 2(-1) + 3(\dfrac{1}{2})\\ \amp= -2 + \dfrac{3}{2} =\dfrac{-1}{2} \end{aligned} \end{equation*}
  2. This expression includes the product of two trig ratios \(6(\tan X)(\cos Y)\text{,}\) where the parentheses indicate multiplication.
    \begin{equation*} \begin{aligned}[t] 6\tan X \cos Y\amp = 6(\tan \alert{30\degree})(\cos \alert{135\degree})\\ \amp= 6(\frac{1}{\sqrt{3}})(\dfrac{-1}{\sqrt{2}}) =\dfrac{-6}{\sqrt{6}} = -\sqrt{6} \end{aligned} \end{equation*}
Checkpoint 5.2

Evaluate each expression for \(X = 30\degree\text{,}\) \(Y = 60\degree\text{.}\)

  1. \(4\sin (3X + 45\degree)\)
  2. \(1 - \cos(4Y)\)
Answer
  1. \(2\sqrt{2}\)
  2. \(\dfrac{3}{2}\)
Caution 5.3

In the previous Exercise, \(\sin (3X + 45\degree)\) is not equal to \(\sin 3X + \sin 45\degree\text{.}\) That is,

\begin{equation*} \sin (90\degree + 45\degree) \not= \sin 90\degree + \sin 45\degree \end{equation*}

(You can check this for yourself.) We must follow the order of operations and evaluate the expression \(3X + 45\degree\) inside parentheses before applying the sine function.

Subsection Simplifying Trigonometric Expressions

When we simplify an algebraic expression, we obtain a new expression that has the same values as the old one, but is easier to work with. For example, we can apply the distributive law and combine like terms to simplify

\begin{equation*} \begin{aligned}[t] 2x(x - 6) + 3(x + 2) \amp = 2x^2 - 12x + 3x + 6\\ \amp = 2x^2 - 9x + 6\\ \end{aligned} \end{equation*}

The new expression is equivalent to the old one, that is, the expressions have the same value when we evaluate them at any value of \(x\text{.}\) For instance, you can check that, at \(x = \alert{3}\text{,}\)

\begin{equation*} \begin{aligned}[t] 2(\alert{3})(\alert{3} - 6) + 3(\alert{3} + 2)\amp = 6(-3) + 3(5) = -3\\ \text{and}~~~~~~~~~~~~~~~ 2(\alert{3})^2 - 9(\alert{3}) + 6 \amp = 18 - 27 + 6 = -3\\ \end{aligned} \end{equation*}

To simplify an expression containing trig ratios, we treat each ratio as a single variable. Compare the two calculations below:

\begin{equation*} \begin{aligned}[t] 8xy - 6xy \amp = 2xy\\ 8\cos \theta \sin \theta - 6\cos\theta \sin \theta \amp = 2\cos \theta \sin \theta\\ \end{aligned} \end{equation*}

Both calculations are examples of combining like terms. In the second calculation, we treat \(\cos \theta\) and \(\sin \theta\) as variables, just as we treat \(x\) and \(y\) in the first calculation.

Example 5.4

Simplify.

  1. \(3\tan A + 4\tan A - 2\cos A\)
  2. \(2 - \sin B + 2\sin B\)
Solution
  1. Combine like terms.
    \begin{equation*} 3\tan A + 4\tan A - 2\cos A = 7 \tan A - 2\cos A \end{equation*}
    Note that \(\tan A\) and \(\cos A\) are not like terms.
  2. Combine like terms.
    \begin{equation*} 2 - \sin B + 2\sin B = 2 + \sin B \end{equation*}
    Note that \(-\sin B\) means \(-1\cdot \sin B\text{.}\)
Checkpoint 5.5

Simplify \(2 \cos t - 4 \cos w \sin w + 3 \cos t - 2 \cos w\)

Answer

\(5\cos t - 4\cos w \sin w - 2 \cos w\)

Caution 5.6

In the previous Exercise, note that \(\cos t\) and \(\cos w\) are not like terms. (We can choose values for \(t\) and \(w\) so that \(\cos t\) and \(\cos w\) have different values.)

Example 5.7

Simplify, and evaluate for \(z = 40\degree\text{.}\)

\begin{equation*} 3 \sin z - \sin z \tan z + 3 \sin z \end{equation*}
Solution

We can combine like terms to get

\begin{equation*} 6 \sin z - \sin z \tan z \end{equation*}

Because \(40\degree\) is not one of the angles for which we know exact trig values, we use a calculator to evaluate the expression.

\begin{equation*} \begin{aligned}[t] 6 \sin 40\degree - (\sin 40\degree)(\tan 40\degree) \amp = 6 (0.6428)(0.8391)\\ \amp = 3.3174\\ \end{aligned} \end{equation*}
Checkpoint 5.8

Simplify, and evaluate for \(x = 25\degree,~y = 70\degree\)

\begin{equation*} 3 \cos x + \cos y - 2 \cos y + \cos x \end{equation*}
Answer

\(3.2832\)

Subsection Powers of Trigonometric Ratios

Compare the two expressions

\begin{equation*} (\cos \theta)^2~~\text{and}~~~ \cos (\theta^2) \end{equation*}

They are not the same.

  • The first expression, \((\cos \theta)^2\text{,}\) says to compute \(\cos \theta\) and then square the result.
  • \(\cos (\theta^2)\) says to square the angle first, and then compute the cosine.

For example, if \(\theta = 30\degree\text{,}\) then

\begin{equation*} \begin{aligned}[t] (\cos 30\degree)^2 \amp = (\dfrac{\sqrt{3}}{2})^2 = \dfrac{3}{4}\\ \text{but}~~~~~~~~ \cos (30^2)\degree\amp =\cos 900\degree = \cos 180\degree = -1\\ \end{aligned} \end{equation*}

We usually write \(\cos^2 \theta\) instead of \((\cos \theta)^2\text{,}\) and \(\cos \theta^2\) for \(\cos (\theta^2)\text{.}\) You must remember that

\begin{equation*} \cos^2 \theta ~~~~ \text{means} ~~~~~(\cos \theta)^2 \end{equation*}

The same notation applies to the other trig ratios, so that

\begin{equation*} \sin^2 \theta = (\sin \theta)^2 ~~~\text{and}~~~ \tan^2 \theta = (\tan \theta)^2 \end{equation*}
Example 5.9

Evaluate \(\sin^2 45\degree\text{.}\)

Solution

\(\sin^2 45\degree = (\sin 45\degree)^2 = (\dfrac{1}{\sqrt{2}})^2 = \dfrac{1}{2}\)

Other powers are written in the same fashion. Thus, for example, \(\sin^3\theta = (\sin \theta)^3\text{.}\)

Checkpoint 5.10

Evaluate \(\tan^4 60\degree\)

Answer

\(9\)

Subsection Products

We can multiply together trigonometric expressions, just as we multiply algebraic expressions. Recall that we use the distributive law in computing products such as

\begin{equation*} x(3x - 2) = 3x^2 - 2x \end{equation*}

and

\begin{equation*} (x - 3)(x + 5) = x^2 + 2x - 15 \end{equation*}
Example 5.11

Using the distributive law, multiply \(~~\cos t (3 \cos t - 2)\text{.}\)

Solution

Think of \(\cos t\) as a single variable, and multiply by each term inside parentheses. (The algebraic form of the calculation is shown on the right in blue).

\begin{equation*} \begin{aligned}[t] \cos t (3 \cos t - 2)\amp = (\cos t)(3 \cos t) - (\cos t)\cdot 2 \amp \amp \blert{x (3x-2) = x\cdot 3x - x\cdot 2}\\ \amp = 3\cos^2 t - 2\cos t \amp \amp \blert{=3x^2 - 2x}\\ \end{aligned} \end{equation*}

Notice that we write \((\cos t)^2\) as \(\cos^2 t\text{.}\)

Checkpoint 5.12

Multiply \(~~2\tan \beta (4 \tan^2 \beta + \tan \alpha)\)

Answer

\(8 \tan^3 \beta + 2 \tan \beta \tan \alpha\)

We can also use the distributive law to multiply binomials that include trig ratios. You may have used the acronym \(\blert{FOIL}\) to remember the four multiplications in a product of binomials: \(\blert{F}\)irst terms, \(\blert{O}\)utside terms, \(\blert{I}\)nside terms, and \(\blert{L}\)ast terms.

Example 5.13

Multiply \(~~(4\sin C - 1)(3 \sin C + 2)\text{.}\)

Solution

This calculation is similar to the product \((4x - 1)(3x+2)\text{,}\) except that the variable \(x\) has been replaced by \(\sin C\text{.}\) Compare the calculations for the two products; first the familiar algebraic product:

\begin{equation*} \begin{aligned}[t] \blert{(4x - 1)(3x + 2)} \amp \blert{= 4x\cdot 3x + 4x\cdot 2 - 1\cdot 3x - 1\cdot 2}\\ \amp \blert{= 12x^2 + 8x - 3x - 2 = 12x^2 + 5x - 2}\\ \end{aligned} \end{equation*}

We compute the product in this example in the same way, but replacing \(x\) by \(\sin C\text{.}\)

\begin{equation*} \begin{aligned}[t] (4\sin C - 1)(\sin C + 2) \amp = (4\sin C)(3\sin C) + (4\sin C)\cdot 2 - 1(3\sin C) - 1\cdot 2\\ \amp = 12\sin^2 C + 5\sin C - 2\\ \end{aligned} \end{equation*}
Checkpoint 5.14

Expand \(~~(4\cos \alpha + 3)^2\)

Answer

\(16 \cos^2 \alpha + 24 \cos \alpha + 9\)

Subsection Factoring

We can factor trigonometric expressions with the same techniques we use for algebraic expressions. In the next two Examples, compare the familiar algebraic factoring with a similar trigonometric expression.

Example 5.15

Factor.

  1. \(6w^2 - 9w\)
  2. \(6 \sin^2 \theta - 9\sin \theta\)
Solution
  1. We factor out the common factor, \(3w\text{.}\)

    \begin{equation*} 6w^2 - 9w = 3w(2w-3) \end{equation*}
  2. We factor out the common factor, \(3\sin \theta\text{.}\)

    \begin{equation*} 6\sin^2 \theta- 9\sin \theta = 3\sin \theta(2\sin \theta-3) \end{equation*}
Checkpoint 5.16

Factor.

  1. \(2a^2 - ab\)
  2. \(2 \cos^2 \phi - \cos \phi \sin \phi\)
Answer
  1. \(a(2a - b)\)

  2. \(\cos \phi (2\cos \phi - \sin \phi)\)

We can also factor quadratic trinomials.

Example 5.17

Factor.

  1. \(t^2 - 3t - 10\)
  2. \(\tan^2 \alpha - 3\tan \alpha - 10\)
Solution
  1. We look for numbers \(p\) and \(q\) so that \((t + p)(t + q) = t^2 - 3t - 10\text{.}\) Their product is \(pq = -10\text{,}\) and their sum is \(p + q = -3\text{.}\) By checking the factors of \(-10\) for the correct sum, we find \(p = -5\) and \(q = 2\text{.}\) Thus,

    \begin{equation*} t^2 - 3t - 10 = (t - 5)(t + 2) \end{equation*}
  2. Now replace \(t\) by \(\tan \alpha\) to find

    \begin{equation*} \tan^2 \alpha - 3\tan \alpha - 10 = (\tan \alpha - 5)(\tan \alpha + 2) \end{equation*}
Checkpoint 5.18

Factor.

  1. \(3z^2 - 2z - 1\)
  2. \(3 \sin^2 \beta - 2\sin \beta - 1\)
Answer
  1. \((3z + 1)(z - 1)\)

  2. \((3\sin \beta + 1)(\sin \beta - 1)\)

Subsection Algebra Refresher

Subsubsection Exercises

Factor.

1

\(2x^2 + 5x - 12\)

2

\(3x^2 - 2x - 8\)

3

\(12x^2 + x - 1\)

4

\(6x^2 - 13x - 15\)

5

\(6x^2 - x - 12\)

6

\(24x^2 + 10x - 21\)

Subsubsection Algebra Refresher Answers

  1. \((2x - 3)(x + 4)\)
  2. \((3x + 4)(x - 2)\)
  3. \((4x - 1)(3x + 1)\)
  4. \((6x + 5)(x - 3)\)
  5. \((2x - 3)(3x + 4)\)
  6. \((6x + 7)(4x - 3)\)

Subsection Section 5.1 Summary

Subsubsection Vocabulary

  • Expression
  • Evaluate
  • Binomial
  • Trinomial
  • Simplify
  • Equivalent expression
  • Like terms
  • Distributive law
  • Factor

Subsubsection Concepts

  1. Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions. To simplify an expression containing trig ratios, we treat each ratio as a single variable.
  2. \(\sin (X + Y)\) is not equal to \(\sin X + \sin Y\) (and the same holds for the other trig ratios). Remember that the parentheses indicate function notation, not multiplication.
  3. We write \(\cos^2 \theta\) to denote \((\cos \theta)^2\text{,}\) and \(\cos^n \theta\) to denote \((\cos \theta)^n\text{.}\) (Similarly for the other trig ratios.)
  4. We can factor trigonometric expressions with the same techniques we use for algebraic expressions.

Subsubsection Study Questions

  1. To evaluate \(\cos^2 30\degree\text{,}\) Delbert used the keystrokes
    \begin{equation*} \boxed {\text{COS}}~~ 30~~ \boxed{x^2}~~ \boxed{\text{ENTER}} \end{equation*}
    and got the answer 1. Were his keystrokes correct? Why or why not?
  2. Make up an example to show that \(\tan (\theta + \phi) \not= \tan \theta + \tan \phi\text{.}\)
  3. Factor each expression, if possible.

    1. \(x^2 - 4x\)
    2. \(x^2 - 4\)
    3. \(x^2 + 4\)
    4. \(x^2 + 4x + 4\)
    5. \(x^2 - 4x + 4\)
    6. \(x^2 + 4x\)
    7. \(x^2 - 4x - 4\)
    8. \(-x^2 + 4\)

Subsubsection Skills

  1. Evaluate trigonometric expressions #1-22
  2. Simplify trigonometric expressions #23-34
  3. Recognize equivalent expressions #35-44
  4. Multiply or expand trigonometric expressions #45-56
  5. Factor trigonometric expressions #57-70

Subsection Homework 5.1

For Problems 1–8, evaluate the expressions, using exact values for the trigonometric ratios.

1

\(5 \tan 135\degree + 6 \cos 60\degree\)

2

\(3 \tan 240\degree + 8 \sin 300\degree\)

3

\(\sin (15\degree + 30\degree)\)

4

\(\cos(2 \cdot 75\degree)\)

5

\(8 \cos^2 (30\degree)\)

6

\(12 \sin^2 (315\degree)\)

7

\(3 \tan^2 (150\degree) - \sin^2 (45\degree)\)

8

\(1 + \tan^2 (120\degree)\)

For Problems 9–16, evaluate the expressions for \(x = 30\degree,~y = 45\degree\text{,}\) and \(z = 60\degree\text{.}\) Give exact values for your answers.

9

\(3 \sin x + 5 \cos y\)

10

\(4 \tan y + 6 \cos y\)

11

\(-2 \tan 3y\)

12

\(\sin (3x - 2x)\)

13

\(\cos^2 x + \sin^2 x\)

14

\(7 \sin^2 y + 7 \cos^2 y\)

15

\(\cos x \cos x - \sin x \sin z\)

16

\(\tan (180\degree - x) \tan x\)

For Problems 17–22,evaluate the expressions using a calculator.

17
  1. \(\sin (10\degree + 40\degree)\)
  2. \(\sin 10\degree + \sin 40\degree\)
  3. \(\sin 10\degree \cos 40\degree + \cos 10\degree \sin 40\degree\)
18
  1. \(\cos (20\degree + 50\degree)\)
  2. \(\cos 20\degree + \cos 50\degree\)
  3. \(\cos 20\degree \cos 50\degree - \sin 20\degree \sin 50\degree\)
19
  1. \(\cos (2 \cdot 24\degree)\)
  2. \(2 \cos 24\degree\)
  3. \(2 \cos^2 (24\degree) - 1\)
20
  1. \(\cos (3 \cdot 49\degree)\)
  2. \(3 \cos 49\degree\)
  3. \(4 \cos^3 (49\degree) - 3\cos 49\degree\)
21
  1. \(\cos^2 (17\degree) + \sin^2 (17\degree)\)
  2. \(\cos^2 (86\degree) + \sin^2 (86\degree)\)
  3. \(\cos^2 (111\degree) + \sin^2 (111\degree)\)
22
  1. \(\dfrac{1}{\cos^2 (25\degree)} - \tan^2 (25\degree)\)
  2. \(\dfrac{1}{\cos^2 (100\degree)} - \tan^2 (100\degree)\)
  3. \(\dfrac{1}{\cos^2 (8\degree)} - \tan^2 (8\degree)\)

For Problems 23–28, combine like terms.

23
  1. \(3x^2 - x - 5x^2\)
  2. \(3\cos^2 \theta - \cos \theta - 5\cos^2 \theta\)
24
  1. \(4x + 5y - y\)
  2. \(4\cos \theta + 5\sin \theta - \sin \theta\)
25
  1. \(-3SC + 7SC\)
  2. \(-3\sin \theta \cos \theta + 7\sin \theta \cos \theta\)
26
  1. \(4SC + 11SC - 17SC\)
  2. \(4\sin \theta \cos \theta + 11\sin \theta \cos \theta - 17\sin \theta \cos \theta\)
27
  1. \(-C^2S^3 + 6C^2S^3\)
  2. \(-\cos^2 \theta \sin^3 \theta + 6 \cos^2 \theta \sin^3 \theta\)
28
  1. \(7C^2S^2 - (2CS)^2\)
  2. \(7\cos^2 \theta \sin^2 \theta - (2\cos \theta \sin \theta)^2\)

For Problems 29–34, simplify the expression, then evaluate.

29

\(\cos t + 2 \cos t \sin t - 3 \cos t,\) for \(t = 143\degree\)

30

\(11 \tan w - 4 \tan w + 6 \tan w \cos w,\) for \(w = 8\degree\)

31

\(7 \tan \theta - 4 \tan \phi + 3 \tan \phi - 6 \tan \theta,\) for \(\theta = 21 \degree, ~ \phi = 89\degree\)

32

\(5 \sin A + \sin B - 6\sin B + 6 \sin A,\) for \(A = 111\degree,~ B = 26\degree\)

33

\(-\sin x \cos x - \sin (2x) + 3\sin x \cos x,\) for \(x = 107\degree\)

34

\(4 \cos u \sin u + 3 \sin 2u - 10 \sin u \cos u,\) for \(u = 2\degree\)

For Problems 35–44, decide whether or not the expressions are equivalent. Explain.

35

\(\cos (x + y),~~\cos x + \cos y\)

36

\(\sin (\theta - phi),~~\sin \theta - \sin \phi\)

37

\(\tan (2A),~~ 2 \tan A\)

38

\(\cos (\frac{1}{2}\beta),~~\frac{1}{2}\cos \beta\)

39

\((\sin \alpha)^2,~~\sin^2 \alpha\)

40

\((\tan B)^2,~~ \tan (B^2)\)

41

\(\sin 3t + \sin 5t,~~\sin 8t\)

42

\(3\cos 2x + 5\cos 2x,~~ 8\cos 2x\)

43

\(\tan (\theta + 45\degree) - \tan \theta,~~ \tan 45\degree\)

44

\(\sin (30\degree + z) + \sin (30\degree - z),~~ \sin 60\degree\)

For Problems 45–56, multiply or expand.

45
  1. \(x(2x-1)\)
  2. \(\sin A (2\sin A - 1)\)
46
  1. \(q(5r-q)\)
  2. \(\cos \theta (5\sin \theta - \cos \theta)\)
47
  1. \(a(b - 3a)\)
  2. \(\tan A (\tan B - 3 \tan A)\)
48
  1. \(3w(2w - z)\)
  2. \(3 \sin \alpha (2 \sin \alpha - \sin \beta)\)
49
  1. \((C + 1)(2C - 1)\)
  2. \((\cos \phi + 1)(2\cos \phi - 1)\)
50
  1. \((3S - 2)(S + 1)\)
  2. \((3\sin B - 2)(\sin B + 1)\)
51
  1. \((a + b)(a - b)\)
  2. \((\cos \theta + \cos \phi)(\cos \theta - \cos \phi)\)
52
  1. \((t - w)(t + 4w)\)
  2. \((\tan \alpha - \tan \beta)(\tan \alpha + 4 \tan \beta)\)
53
  1. \((1 - T)^2\)
  2. \((1 - \tan \theta)^2\)
54
  1. \((2 + 3S)^2\)
  2. \((2 + 3\sin \theta)^2\)
55
  1. \((T^2 + 2)(T^2 - 2)\)
  2. \((\tan^2 \theta + 2)(\tan^2 \theta - 2)\)
56
  1. \((2c^2 - 3)(2c^2 + 3)\)
  2. \((2\cos^2\phi - 3)(2\cos^2 \phi + 3)\)

For Problems 57–70, factor.

57
  1. \(9m + 15n\)
  2. \(9\cos\alpha + 15\cos\beta\)
58
  1. \(12p - 20q\)
  2. \(12\sin \theta - 20 \sin \phi\)
59
  1. \(5r^2 - 10 qr\)
  2. \(5\tan^2 C - 10 \tan B \tan C\)
60
  1. \(2x^2 - 6xy\)
  2. \(2\sin^2 a - 6\sin A \cos A\)
61
  1. \(9C^2 - 1\)
  2. \(9\cos^2 \beta - 1\)
62
  1. \(25S^2 - 16S\)
  2. \(25\sin^2 \beta - 16 \sin \beta\)
63
  1. \(6T^3 - 8T^2\)
  2. \(6\tan^3 A - 8 \tan^2 A\)
64
  1. \(9T^2 - 15T^3\)
  2. \(9\tan^2 B - 15\tan^3 B\)
65
  1. \(t^2 - t - 20\)
  2. \(\tan^2 \theta - \tan \theta - 20\)
66
  1. \(s^2 + 5s - 6\)
  2. \(\sin^2 A + 5 \sin A - 6\)
67
  1. \(3c^2 + 2c - 1\)
  2. \(3\cos^2 B + 2\cos B - 1\)
68
  1. \(8S^2 - 6S + 1\)
  2. \(8\sin^2 \phi - 6 \sin \phi + 1\)
69
  1. \(6S^2 - 5S - 1\)
  2. \(6\sin^2 \alpha - 5\sin \alpha - 1\)
70
  1. \(T^2 - 4T - 12\)
  2. \(\tan^2 \alpha - 4\tan \alpha - 12\)