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Section 10.5 Chapter Summary and Review
Subsection Key Concepts
Polar Coordinates. The polar coordinates of a point \(P\) in the plane are \((r, \theta)\text{,}\) where
\(\abs{r}\) is the distance from to the pole,
\(\theta\) is the angle measured counterclockwise from the polar axis to the ray through \(P\) from the pole.
Non-Uniqueness of Polar Coordinates.
Any point with polar coordinates \((r, \theta)\) also has coordinates \((r, \theta + 2k\pi)\text{,}\) where \(k\) is an integer.
The point \((r, \theta)\) can also be designated by \((-r, \theta + \pi)\text{.}\)
The pole has coordinates \((0, \theta)\text{,}\) for any value of \(\theta\text{.}\)
In the polar plane, the coordinate grid lines are circles centered at the pole, with equations \(r=k\text{,}\) and lines through the pole, with equations \(\theta = k\text{.}\)
Conversion Equations.
To convert from polar coordinates
\((r, \theta)\) to Cartesian:
\begin{align*}
x \amp = r\cos (\theta)\\
y \amp = r \sin (\theta)
\end{align*}
To convert from Cartesian coordinates
\((x,y)\) to polar:
\begin{align*}
\amp r = \sqrt{x^2+y^2}\\
\amp \tan (\theta) = \dfrac{y}{x}
\end{align*}
where the choice of \(\theta\) depends on the quadrant.
To convert an equation from Cartesian to polar coordinates, we replace each \(x\) with \(r\cos (\theta)\) and each with \(y\) with \(r\sin)\text{.}\) To convert an equation from polar to Cartesian coordinates, look for expressions of the form \(r\cos (\theta),~r\sin (\theta),~r^2\text{,}\) or \(\tan (\theta)\text{.}\)
When graphing an equation in polar coordinates, we think of sweeping around the pole in the counterclockwise direction, and at each angle \(\theta\) the \(r\) -value tells us how far the graph is from the pole.
Standard graphs in polar coordinates include circles and roses, cardioids and limaçons, lemniscates, and spirals.
To find the intersection points of the polar graphs \(r=f(\theta)\) and \(r=g(\theta)\) we solve the equation \(f(\theta)=g(\theta)\text{.}\) In addition, we should always check whether the pole is a point on both graphs.
Imaginary Unit.
We define the imaginary unit, \(i\)
\begin{equation*}
i^2=-1~~~~~~\text{or}~~~~~~i=\sqrt{-1}
\end{equation*}
The square root of a negative number is an imaginary number: if \(a \gt 0,~ \sqrt{-a}=i\sqrt{a}\)
A complex number \(z\) is the sum of a real number and an imaginary number, \(z=a+bi\text{.}\)
We can perform the four arithmetic operations on complex numbers.
Operations on Complex Numbers.
\begin{equation*}
z_1+z_2=(a+bi)+(c+di)=(a+c)+(b+d)i
\end{equation*}
\begin{equation*}
z_1-z_2=(a+bi)-(c+di)=(a-c)+(b-d)i
\end{equation*}
\begin{equation*}
z_1z_2=(a+bi)(c+di) = (ac-bd)+(ad+bc)i
\end{equation*}
\begin{equation*}
\dfrac{z_1}{z_2} = \dfrac{a+bi}{c+di} = \dfrac{a+bi}{c+di} \cdot \dfrac{c-di}{c-di} = \dfrac{ac+bd}{c^2+d^2} + \dfrac{bc-ad}{c^2+d^2}i
\end{equation*}
The product of a nonzero complex number and its conjugate is always a positive real number.
\begin{equation*}
z \bar{z} = (a+bi)(a-bi) = a^2 - b^2i^2 = a^2 - b^2(-1)=a^2+b^2\text{.}
\end{equation*}
We can graph complex numbers in the complex plane.
We can visualize the sum of two complex numbers by vector addition in the complex plane.
Fundamental Theorem of Algebra. Let \(p(x)\) be a polynomial of degree \(n \ge 1\text{.}\) Then \(p(x)\) has exactly \(n\) complex zeros.
The nonreal zeros of a polynomial with real coefficients always occur in conjugate pairs.
Multiplying a complex number by \(i\) rotates its graph by \(90\degree\) around the origin.
Polar Form for a Complex Number.
The complex number \(z=a+bi\) can be written in the polar form
\begin{equation*}
z=r(\cos(\theta)+i\sin (\theta))
\end{equation*}
where
\begin{equation*}
r=\sqrt{a^2+b^2}
\end{equation*}
and \(\theta\) is defined by
\begin{equation*}
a=r \cos (\theta),~~~~b=r\sin (\theta),~~~~0 \le \theta \le 2\pi
\end{equation*}
The angle \(\theta\) is called the argument of the complex number, and \(r\) is its length, or modulus .
Product and Quotient in Polar Form.
If \(z_1=r(\cos \alpha+i\sin \alpha)\) and \(z_2=R(\cos \beta+i\sin \beta)\text{,}\) then
\begin{equation*}
z_1z_2=rR(\cos (\alpha + \beta) + i \sin (\alpha + \beta))
\end{equation*}
and
\begin{equation*}
\dfrac{z_1}{z_2}=\dfrac{r}{R}(\cos (\alpha - \beta) + i \sin (\alpha - \beta))
\end{equation*}
De Moivre’s Theorem.
If \(z=r(\cos(\alpha)+i\sin( \alpha))\) is a complex number in polar form, and \(n\) is a positive integer, then
\begin{equation*}
z^n= r^n(\cos (n\alpha)+i\sin (n\alpha))
\end{equation*}
Roots of a Complex Number.
A complex number \(z=r(\cos (\alpha) + i\sin (\alpha))\) in polar form has \(n\) complex \(n\) th roots, given by
\begin{equation*}
\blert{z_k = r^{1/n}\left(\cos \left(\dfrac{\alpha + 2\pi k}{n}\right) + i\sin \left(\dfrac{\alpha + 2\pi k}{n}\right)\right)}
\end{equation*}
for \(k = 0,~1,~2, \cdots,~ n-1\text{.}\)
Exercises Review Problems
Exercise Group.
For Problems 1–4, use the grid at right to plot the points whose polar coordinates are given.
1. \(\left(4, \dfrac{5\pi}{6}\right)\)
2. \(\left(1, \dfrac{-5\pi}{3}\right)\)
3. \(\left(-3, \dfrac{\pi}{2}\right)\)
4. \(\left(2, \dfrac{7\pi}{4}\right)\)
Exercise Group. For Problems 5–8, convert the polar coordinates to Cartesian coordinates.
5. \(\left(1, \dfrac{5\pi}{4}\right)\)
6. \(\left(0, \dfrac{\pi}{12}\right)\)
7. \((3.4, -1.5)\)
8. \((-5.6, -1.1)\)
Exercise Group. For Problems 9–12, convert the Cartesian coordinates to polar coordinates with \(r \ge 0\) and \(0 \le \theta \le 2\pi\text{.}\) Give exact values for \(r\) and \(\theta\text{.}\)
9. \((-3, 3)\)
10. \((0, -2)\)
11. \((5, -2)\)
12. \((-15, -8)\)
Exercise Group. For Problems 13–16, sketch the region described by the inequalities.
13. \(r \ge 0,~ \dfrac{-\pi}{4} \le \theta \le \dfrac{\pi}{4}\)
14. \(1 \le r \le 3,~0 \le \theta \le \pi\)
15. \(0 \le r \le 2\)
16. \(r \ge 4\)
Exercise Group. For Problems 17–20, convert the equation into Cartesian coordinates.
17. \(r=1\)
18. \(r=-3\sec (\theta)\)
19. \(r=\dfrac{6}{1-2\cos(\theta)}\)
20. \(3\tan (\theta) = 6r\sin (\theta) - 1\)
Exercise Group. For Problems 21–24, convert the equation into polar coordinates.
21. \(x+y = 2\)
22. \(\sqrt{x^2+y^2} = 4y\)
23. \(\dfrac{y}{x}=\sqrt{x^2+y^2}\)
24. \(y^2 = 2y = x - x^2\)
Exercise Group. For Problems 25–28, use the catalog of polar graphs to help you identify and sketch the curve. Check your work by graphing with a calculator.
25. \(r=3\)
26. \(\theta = \dfrac{3\pi}{4}\)
27. \(r=6\cos (\theta)\)
28. \(r^2=9\sin (2\theta)\)
Exercise Group. For Problems 29–32, write a polar equation for the graph.
Exercise Group. For Problems 33–36, find the coordinates of the intersection points of the two curves analytically. Then graph the curves to verify your answers.
33. \(r=3+2\sin (\theta),~r=4\)
34. \(r=3\cos(\theta),~r=\sqrt{3}\sin (\theta)\)
35. \(r=4\sin (\theta),~r=-4\cos (\theta)\)
36. \(r=2+6\sin (\theta),~r=4\sin (\theta)\)
Exercise Group. For Problems 37–40, perform the indicated operations on the complex numbers.
37. \(\dfrac{5-10i}{2-i}\)
38. \((4-7i)(1+i)\)
39. \(5i(2-i)-(7+6i)\)
40. \(-8+3i+\dfrac{9-4i}{i}\)
Exercise Group. For Problems 41–44, evaluate the polynomial for the given values of the variable.
41.
\(z^2+4z+6\)
\(\displaystyle z=-2+i\)
\(\displaystyle z=-2-1\)
42.
\(z^2-6z+12\)
\(\displaystyle z=3-2i\)
\(\displaystyle z=3+21\)
43.
\(3w^2-18w+31\)
\(\displaystyle w=3+4i\)
\(\displaystyle w=3-4i\)
44.
\(2w^2+8w+11\)
\(\displaystyle w=-2-5i\)
\(\displaystyle w=-2+51\)
45. Verify that \(z_1=2+i\) and \(z_2=2-i\) are roots of the equation \(x^2-4x+5=0\text{.}\)
46. Verify that \(z_1=-3+4i\) and \(z_2=-3-4i\) are roots of the equation \(x^2+6x+25=0\text{.}\)
Exercise Group. For Problems 47–50, expand the product of polynomials.
47. \([z-(-2+i)][z-(-2-i)]\)
48. \([w-(1+3i)][w-(1-3i)])]\)
49. \([s+(5+4i)][s+(5-4i)]\)
50. \([x+(-6+i)][x+(-6-i)]\)
Exercise Group. For Problems 51–52, sketch the set of points in the complex plane.
51.
\(z_1=-3+2i,~\) \(z_2=-3-2i,~\) \(z_3=3+2i,~\) \(z_4=3-2i\)
52.
\(w_1=4-6i,~\) \(w_2=4+6i,~\) \(w_3=-4-6i,~\) \(w_4=-4+6i\)
Exercise Group.
For Problems 53–56,
Given one solution of a quadratic equation with rational coefficients, find the other solution.
Write a quadratic equation that has those solutions.
53. \(-1+7i\)
54. \(2-5i\)
55. \(3-\sqrt{2}i\)
56. \(4+\sqrt{3}i\)
Exercise Group. For Problems 57–60, write the complex numbers in standard form. Give exact values for your answers.
57. \(10\left(\cos \left(\dfrac{-\pi}{6}\right) + i\sin \left(\dfrac{-\pi}{6}\right)\right)\)
58. \(8\left(\cos \left(\dfrac{5\pi}{4}\right) + i\sin \left(\dfrac{5\pi}{4}\right)\right)\)
59. \(5\sqrt{2}\left(\cos \left(\dfrac{\pi}{4}\right) + i\sin \left(\dfrac{\pi}{4}\right)\right)\)
60. \(6\sqrt{3}\left(\cos \left(\dfrac{-\pi}{3}\right) + i\sin \left(\dfrac{-\pi}{3}\right)\right)\)
Exercise Group. For Problems 61–66, write the complex numbers in polar form. Give exact values for your answers.
61. \(3-3i\)
62. \(-4-4i\)
63. \(-5\)
64. \(-7i\)
65. \(-1-\sqrt{3}i\)
66. \(6+2\sqrt{3}i\)
Exercise Group. For Problems 67–70, find the product \(z_1z_2\) and the quotient \(\dfrac{z_1}{z_2}\text{.}\)
67.
\(z_1=8\left(\cos \left(\dfrac{\pi}{6}\right) + i\sin \left(\dfrac{\pi}{6}\right)\right)\) \(z_2=2\left(\cos \left(\dfrac{5\pi}{6}\right) + i\sin \left(\dfrac{5\pi}{6}\right)\right)\)
68.
\(z_1=9\left(\cos \left(\dfrac{-2\pi}{3}\right) + i\sin \left(\dfrac{-2\pi}{3}\right)\right)\) \(z_2=3\left(\cos \left(\dfrac{\pi}{3}\right) + i\sin \left(\dfrac{\pi}{3}\right)\right)\)
69.
\(z_1=5\left(\cos \left(\dfrac{-7\pi}{12}\right) + i\sin \left(\dfrac{-7\pi}{12}\right)\right)\) \(z_2=\dfrac{1}{2}\left(\cos\left( \left(\dfrac{\pi}{4}\right) + i\sin \left(\dfrac{\pi}{4}\right)\right)\right)\)
70.
\(z_1=14\left(\cos \left(\dfrac{3\pi}{2}\right) + i\sin \left(\dfrac{3\pi}{2}\right)\right)\) \(z_2=2\left(\cos \dfrac{\pi}{6} + i\sin \dfrac{\pi}{6}\right)\)
Exercise Group. For Problems 71–74, find the power.
71. \(\left(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\right)^{12}\)
72. \(\left(1-\sqrt{3}i\right)^6\)
73. \(\left(-\sqrt{5} + \sqrt{5}i\right)^{-4}\)
74. \((-1-i)^{-8}\)
Exercise Group.
For Problems 75–78,
Find the roots and plot them in the complex plane.
Write the roots in standard form.
75. The square roots of \(-16i\)
76. The cube roots of \(-8\)
77. The cube roots of \(-27i\)
78. The square roots of \(-2+2\sqrt{3}i\)
Exercise Group. For Problems 79–82, solve the equation.
79. \(z^6+27=0\)
80. \(z^4-6z^2+12=0\)
81. \(z^4+2z^2+4=0\)
82. \(z^6-8=0\)
