1.
[latex]\left(4, \dfrac{5\pi}{6}\right)[/latex]
Chapter 10: Polar Coordinates and Complex Numbers
Exercises: Chapter 10 Review Problems
Chapter Review Suggested Problems
Problems: #4, 8, 12, 14, 20, 28, 30, 36, 38, 46, 50, 52, 53, 58, 66, 68, 72, 76, 82
Exercises for Chapter 10 Review
Exercise Group
For Problems 1–4, use the grid at right to plot the points whose polar coordinates are given.
2.
[latex]\left(1, \dfrac{-5\pi}{3}\right)[/latex]
3.
[latex]\left(-3, \dfrac{\pi}{2}\right)[/latex]
4.
[latex]\left(2, \dfrac{7\pi}{4}\right)[/latex]
Exercise Group
For Problems 5–8, convert the polar coordinates to Cartesian coordinates.
5.
[latex]\left(1, \dfrac{5\pi}{4}\right)[/latex]
6.
[latex]\left(0, \dfrac{\pi}{12}\right)[/latex]
7.
[latex](3.4, -1.5)[/latex]
8.
[latex](-5.6, -1.1)[/latex]
Exercise Group
For Problems 9–12, convert the Cartesian coordinates to polar coordinates with [latex]r \ge 0[/latex] and [latex]0 \le \theta \le 2\pi\text{.}[/latex] Give exact values for [latex]r[/latex] and [latex]\theta\text{.}[/latex]
9.
[latex](-3, 3)[/latex]
10.
[latex](0, -2)[/latex]
11.
[latex](5, -2)[/latex]
12.
[latex](-15, -8)[/latex]
Exercise Group
For Problems 13–16, sketch the region described by the inequalities.
13.
[latex]r \ge 0,~ \dfrac{-\pi}{4} \le \theta \le \dfrac{\pi}{4}[/latex]
14.
[latex]1 \le r \le 3,~0 \le \theta \le \pi[/latex]
15.
[latex]0 \le r \le 2[/latex]
16.
[latex]r \ge 4[/latex]
Exercise Group
For Problems 17–20, convert the equation into Cartesian coordinates.
17.
[latex]r=1[/latex]
18.
[latex]r=-3\sec \theta[/latex]
19.
[latex]r=\dfrac{6}{1-2\cos\theta}[/latex]
20.
[latex]3\tan \theta = 6r\sin \theta - 1[/latex]
Exercise Group
For Problems 21–24, convert the equation into polar coordinates.
21.
[latex]x+y = 2[/latex]
22.
[latex]\sqrt{x^2+y^2} = 4y[/latex]
23.
[latex]\dfrac{y}{x}=\sqrt{x^2+y^2}[/latex]
24.
[latex]y^2 = 2y = x - x^2[/latex]
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.
[latex]r=3[/latex]
26.
[latex]\theta = \dfrac{3\pi}{4}[/latex]
27.
[latex]r=6\cos \theta[/latex]
28.
[latex]r^2=9\sin 2\theta[/latex]
Exercise Group
For Problems 29–32, write a polar equation for the graph.
29.
30.
31.
32.
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.
[latex]r=3+2\sin \theta,~r=4[/latex]
34.
[latex]r=3\cos\theta,~r=\sqrt{3}\sin \theta[/latex]
35.
[latex]r=4\sin \theta,~r=-4\cos \theta[/latex]
36.
[latex]r=2+6\sin \theta,~r=4\sin \theta[/latex]
Exercise Group
For Problems 37–40, perform the indicated operations on the complex numbers.
37.
[latex]\dfrac{5-10i}{2-i}[/latex]
38.
[latex](4-7i)(1+i)[/latex]
39.
[latex]5i(2-i)-(7+6i)[/latex]
40.
[latex]-8+3i+\dfrac{9-4i}{i}[/latex]
Exercise Group
For Problems 41–44, evaluate the polynomial for the given values of the variable.
41.
[latex]z^2+4z+6[/latex]
- [latex]\displaystyle z=-2+i[/latex]
- [latex]\displaystyle z=-2-1[/latex]
42.
[latex]z^2-6z+12[/latex]
- [latex]\displaystyle z=3-2i[/latex]
- [latex]\displaystyle z=3+21[/latex]
43.
[latex]3w^2-18w+31[/latex]
- [latex]\displaystyle w=3+4i[/latex]
- [latex]\displaystyle w=3-4i[/latex]
44.
[latex]2w^2+8w+11[/latex]
- [latex]\displaystyle w=-2-5i[/latex]
- [latex]\displaystyle w=-2+51[/latex]
45.
Verify that [latex]z_1=2+i[/latex] and [latex]z_2=2-i[/latex] are roots of the equation [latex]x^2-4x+5=0\text{.}[/latex]
46.
Verify that [latex]z_1=-3+4i[/latex] and [latex]z_2=-3-4i[/latex] are roots of the equation [latex]x^2+6x+25=0\text{.}[/latex]
Exercise Group
For Problems 47–50, expand the product of polynomials.
47.
[latex][z-(-2+i)][z-(-2-i)][/latex]
48.
[latex][w-(1+3i)][w-(1-3i)])][/latex]
49.
[latex][s+(5+4i)][s+(5-4i)][/latex]
50.
[latex][x+(-6+i)][x+(-6-i)][/latex]
Exercise Group
For Problems 51–52, sketch the set of points in the complex plane.
51.
[latex]z_1=-3+2i,~[/latex] [latex]z_2=-3-2i,~[/latex] [latex]z_3=3+2i,~[/latex] [latex]z_4=3-2i[/latex]
52.
[latex]w_1=4-6i,~[/latex] [latex]w_2=4+6i,~[/latex] [latex]w_3=-4-6i,~[/latex] [latex]w_4=-4+6i[/latex]
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.
[latex]-1+7i[/latex]
54.
[latex]2-5i[/latex]
55.
[latex]3-\sqrt{2}i[/latex]
56.
[latex]4+\sqrt{3}i[/latex]
Exercise Group
For Problems 57–60, write the complex numbers in standard form. Give exact values for your answers.
57.
[latex]10(\cos \dfrac{-\pi}{6} + i\sin \dfrac{-\pi}{6})[/latex]
58.
[latex]8(\cos \dfrac{5\pi}{4} + i\sin \dfrac{5\pi}{4})[/latex]
59.
[latex]5\sqrt{2}(\cos \dfrac{\pi}{4} + i\sin \dfrac{\pi}{4})[/latex]
60.
[latex]6\sqrt{3}(\cos \dfrac{-\pi}{3} + i\sin \dfrac{-\pi}{3})[/latex]
Exercise Group
For Problems 61–66, write the complex numbers in polar form. Give exact values for your answers.
61.
[latex]3-3i[/latex]
62.
[latex]-4-4i[/latex]
63.
[latex]-5[/latex]
64.
[latex]-7i[/latex]
65.
[latex]-1-\sqrt{3}i[/latex]
66.
[latex]6+2\sqrt{3}i[/latex]
Exercise Group
For Problems 67–70, find the product [latex]z_1z_2[/latex] and the quotient [latex]\dfrac{z_1}{z_2}\text{.}[/latex]
67.
[latex]z_1=8\left(\cos \dfrac{\pi}{6} + i\sin \dfrac{\pi}{6}\right)[/latex] [latex]z_2=2\left(\cos \dfrac{5\pi}{6} + i\sin \dfrac{5\pi}{6}\right)[/latex]
68.
[latex]z_1=9\left(\cos \dfrac{-2\pi}{3} + i\sin \dfrac{-2\pi}{3}\right)[/latex][latex]z_2=3\left(\cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}\right)[/latex]
69.
[latex]z_1=5\left(\cos \dfrac{-7\pi}{12} + i\sin \dfrac{-7\pi}{12}\right)[/latex][latex]z_2=\dfrac{1}{2}\left(\cos \dfrac{\pi}{4} + i\sin \dfrac{\pi}{4}\right)[/latex]
70.
[latex]z_1=14\left(\cos \dfrac{3\pi}{2} + i\sin \dfrac{3\pi}{2}\right)[/latex][latex]z_2=2\left(\cos \dfrac{\pi}{6} + i\sin \dfrac{\pi}{6}\right)[/latex]
Exercise Group
For Problems 71–74, find the power.
71.
[latex]\left(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\right)^{12}[/latex]
72.
[latex]\left(1-\sqrt{3}i\right)^6[/latex]
73.
[latex]\left(-\sqrt{5} + \sqrt{5}i\right)^{-4}[/latex]
74.
[latex](-1-i)^{-8}[/latex]
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 [latex]-16i[/latex]
76.
The cube roots of [latex]-8[/latex]
77.
The cube roots of [latex]-27i[/latex]
78.
The square roots of [latex]-2+2\sqrt{3}i[/latex]
Exercise Group
For Problems 79–82, solve the equation.
79.
[latex]z^6+27=0[/latex]
80.
[latex]z^4-6z^2+12=0[/latex]
81.
[latex]z^4+2z^2+4=0[/latex]
82.
[latex]z^6-8=0[/latex]