11.9: Polar Coordinates and Complex Numbers
- Page ID
- 123034
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)10.1 Polar Coordinates
Homework 10-1
1. 
3. 
5. 
7. 
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.
a \(\left(-2, \dfrac{11 \pi}{6}\right)\)
b \(\left(2, \dfrac{-7 \pi}{6}\right)\)
35.
a \((-3,0)\)
b \((3,-\pi)\)
37.
a \((-2.3,2.06)\)
b \((2.3,-1.08)\)
39. 
41. 
43. 
45. \(r \geq 0, \dfrac{\pi}{6} \leq \theta \leq \dfrac{\pi}{3}\)
47. \(r \geq 1, \dfrac{\pi}{2} \leq \theta \leq \pi\)
49. \(-1 \leq r \leq 1, \dfrac{3 \pi}{4} \leq \theta \leq \pi\)
51. \(x^2+y^2=2\)
53. \(x^2+y^2=4 x\)
55. \(y = 1\)
57. \(y = 2x\)
59. \(x^2+y^2=3 x\)
61. \(x^2=4-4 y\)
63. \(2x + y = 1\)
65. \(r=2 \sec \theta\)
67. \(2 r^2=\sec \theta \csc \theta\)
69. \(r=4 \cot \theta \csc \theta\)
71. \(r = 4\)
73.
\begin{aligned}
d & =\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2} \\
& =\sqrt{\left(r_2 \cos \theta_2-r_1 \cos \theta_1\right)^2+\left(r_2 \sin \theta_2-r_1 \sin \theta_1\right)^2} \\
& =\sqrt{r_2^2 \cos ^2 \theta_2-2 r_2 r_1 \cos \theta_2 \cos \theta_1+r_1^2 \cos ^2 \theta_1+r_2^2 \sin ^2 \theta_2-2 r_2 r_1 \sin \theta_2 \sin \theta_1+r_1^2 \sin ^2 \theta_1} \\
& =\sqrt{r_2^2+r_1^2-2 r_2 r_1\left(\cos \theta_2 \cos \theta_1-\sin \theta_2 \sin \theta_1\right)} \\
&= \sqrt{r_1^2+r_2^2-2r_1r_2\cos(\theta_2 - \theta_1)}
\end{aligned}
10.2 Polar Graphs
Homework 10-2
1.
a 
\(k\) is the radius
b \(x^2 + y^2 = 1, x^2 + y^2 = 4, x^2 + y^2 = 9\)
3.
a 
\(\tan k\) is the slope
b \(y = \dfrac{x}{\sqrt{3}}, y = \sqrt{3} x, y = -\sqrt{3}x, y = \dfrac{x}{\sqrt{3}}\)
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.
a 
b
| \(\theta\) | \(\pi\) | \(\dfrac{5\pi}{4}\) | \(\dfrac{3\pi}{2}\) | \(\dfrac{7\pi}{4}\) | \(2\pi\) |
| \(r\) | -4 | \(-2\sqrt{2}\) | 0 | \(2\sqrt{2}\) | 4 |
The graph is traced again.
c center: (2,0), radius: 2
d \((x-2)^2 + y^2 = 4\)
9.
a 
b For \(a > 0\), \(a\) is the radius of a circle centered on the positive \(y\)-axis; for \(a < 0\), \(|a|\) is the radius of a circle centered on the negative \(y\)-axis.
11.
a 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | 1 | 2 | 1 | 0 | 1 |
b 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | -1 | 0 | -1 | -2 | -1 |
c 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | 1 | 0 | 1 | 2 | 1 |
d 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | -1 | -2 | -1 | 0 | -1 |
13.
a 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | 3 | 2 | 1 | 2 | 3 |
b 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | 1 | 2 | 3 | 2 | 1 |
c 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | 3 | 1 | -1 | 1 | 3 |
d 
| \(\theta\) | 0 | \(\dfrac{\pi}{2}\) | \(\pi\) | \(\dfrac{3\pi}{2}\) | \(2\pi\) |
| \(r\) | -1 | 1 | 3 | 1 | -1 |
15.
a 
There are \(n\) petals if \(n\) is odd, and \(2 n\) petals if \(n\) is even.
b
\(n=2: \dfrac{\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{5 \pi}{4}, \dfrac{7 \pi}{4} ;\)
\(n=3: \dfrac{\pi}{6}, \dfrac{5 \pi}{6}, \dfrac{3 \pi}{2} ;\)
\(n=4: \dfrac{\pi}{8}, \dfrac{3 \pi}{8}, \dfrac{5 \pi}{8}, \dfrac{7 \pi}{8}, \dfrac{9 \pi}{8}, \dfrac{11 \pi}{8}, \dfrac{13 \pi}{8}, \dfrac{15 \pi}{8} ;\)
\(n=5: \dfrac{\pi}{10}, \dfrac{\pi}{2}, \dfrac{9 \pi}{10}, \dfrac{13 \pi}{10}, \dfrac{17 \pi}{10}\)
c. 
\(a\) is the length of the petal.
17.
a \(r = \pm 3 \sqrt{\cos 2 \theta}\)
b 
c \(a\) is the length of the loop.
19. 
21.
a
| \(\theta\) | 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 |
b
| \(\theta\) | 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.
a
| \(\theta\) | 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 |
b
| \(\theta\) | 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 |
25. circle
27. line
29. circle
31. cardioid
33. limaçon
25. rose
37. rose
39. limaçon
41. lemniscate
43. circle
45. arcs of a circle
47. semicircle
49. rose
51. cardioid
53. parabola
55. ellipse
57. 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. \((0,0),\left(\dfrac{1}{2}, \dfrac{\pi}{3}\right),\left(\dfrac{1}{2}, \dfrac{5 \pi}{3}\right)\)
69. \((0,0), \left(\dfrac{3}{\sqrt{2}}, \dfrac{\pi}{4}\right), \left(\dfrac{-3}{\sqrt{2}}, \dfrac{5 \pi}{4}\right)\)
71. \(\left(1, \dfrac{\pi}{2}\right), \left(1, \dfrac{3 \pi}{2}\right)\)
73. \(\left(\dfrac{4+\sqrt{2}}{2}, \dfrac{3 \pi}{4}\right),\left(\dfrac{4-\sqrt{2}}{2}, \dfrac{7 \pi}{4}\right)\)
75. 
77. 
79. 
81. 
83. The curve has \(n\) large loops and \(n\) small loops.
10.3 Complex Numbers
Homework 10-3
1.
a \(5 i-4\)
b \(-4+i\)
c \(\dfrac{-5}{6}-\dfrac{\sqrt{2}}{6} i\)
3. \(-3 \pm 2 i\)
5. \(\dfrac{1}{6} \pm \dfrac{\sqrt{11}}{6} i\)
7. \(13+4 i\)
9. \(-0.8+3.8 i\)
11. \(20-10 i\)
13. \(-14+34 i\)
15. \(46+14 i \sqrt{3}\)
17. 52
19. \(-2-2 i\)
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.
a -1
b 1
c \(-i\)
d -1
35.
a 0
b 0
37.
a 0
b 0
39.
a 0
b 0
41. \(4z^2 + 49\)
43. \(x^2+6x+10\)
45. \(v^2-8c+17\)
47. 
49. 
51. 
53. 
55. 
57. 
59. \((a+b i)(c+d i)=a c+a d i+b c i+b d i^2=(a c-b d)+(a d+b c) i\)
61.
\begin{aligned}
z_1+z_2 &= (a+b i)+(c+d i)=(a+c)+(b+d) i \\
&= (c+a)+(d+b) i=(c+d i)+(a+b i)=z_2+z_1 \\ \\
z_1 z_2 & =(a+b i)(c+d i)=(a c-b d)+(a d+b c) i \\
& =(c a-d b)+(d a+c b) i=z_2 z_1
\end{aligned}
63.
a \(z+\bar{z}=(a+b i)+(a-b i)=2 a ; \quad z-\bar{z}=(a+b i)-(a-b i)=-2 b i\)
b \(z \bar{z}=(a+b i)(a-b i)=a^2+b^2=|z|^2\)
65. No. Let \(t=i\) and \(z=-i\). Then \(w=t+z=i-i=0\), so \(|w|=0\), but \(|t|+|z|=|i|+|-i|=1+1=2\).
67.
a \(2-\sqrt{5}\)
b \(x^2-4 x-1=0\)
69.
a \(4+3 i\)
b \(x^2-8 x+25=0\)
71. \(x^4-6 x^3+23 x^2-50 x+50=0\)
73. \(x^4-7 x^3+20 x^2-19 x+13=0\)
10.4 Polar Form for Complex Numbers
Homework 10-4
1. \(1, i,-1,-i, 1\)
3. \(1+2 i,-2+i\)
5. \(-3+3 i \sqrt{3}\)
7. \(-1+i\)
9. \(2.34-4.21 i\)
11. \(-5.07+10.88 i\)
13. \(3\left(\cos \dfrac{\pi}{2}+i \sin \dfrac{\pi}{2}\right), 3\left(\cos \dfrac{3 \pi}{2}+i \sin \dfrac{3 \pi}{2}\right)\)
15. \(2 \sqrt{3}\left(\cos \dfrac{7 \pi}{6}+i \sin \dfrac{7 \pi}{6}\right), 2 \sqrt{3}\left(\cos \dfrac{11 \pi}{6}+i \sin \dfrac{11 \pi}{6}\right)\)
17. \(4.47(\cos 2.68+i \sin 2.68), \quad 4.47(\cos 5.82+i \sin 5.82)\)
19. \(8.60(\cos 5.78+i \sin 5.78), \quad 8.60(\cos 0.51+i \sin 0.51)\)
21. \(5(\cos 0.93+i \sin 0.93), \quad 5(\cos 5.36+i \sin 5.36), \quad 5(\cos 2.21+i \sin 2.21), \quad 5(\cos 4.07+i \sin 4.07)\)
23. If \(z=r(\cos \theta+i \sin \theta)\), then \(\bar{z}=r(\cos (2 \pi-\theta)+i \sin (2 \pi-\theta)\)
25. \(z_1 z_2=2\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)=\sqrt{3}+i ; \dfrac{z_1}{z_2}=8\left(\cos \dfrac{\pi}{2}+i \sin \dfrac{\pi}{2}\right)=8 i\)
27. \(z_1 z_2=6\left(\cos \dfrac{9 \pi}{10}+i \sin \dfrac{9 \pi}{10}\right); \dfrac{z_1}{z_2}=\dfrac{3}{2}\left(\cos \dfrac{3 \pi}{10}+i \sin \dfrac{3 \pi}{10}\right)\)
29. \(z_1 z_2=8 ; \dfrac{z_1}{z_2}=\dfrac{1}{2}\)
31. \(z_1 z_2=4 \sqrt{2}\left(\cos \dfrac{7 \pi}{12}+i \sin \dfrac{7 \pi}{12}\right) ;\dfrac{z_1}{z_2}=2 \sqrt{2}\left(\cos \dfrac{13 \pi}{12}+i \sin \dfrac{13 \pi}{12}\right)\)
33. \(-128 - 128 i\)
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.
a \(3\left(\cos \dfrac{\pi}{4}+i \sin \dfrac{\pi}{4}\right), 3\left(\cos \dfrac{3 \pi}{4}+i \sin \dfrac{3 \pi}{4}\right)\)
b \(\dfrac{3}{\sqrt{2}}+\dfrac{3}{\sqrt{2}} i, \quad \dfrac{-3}{\sqrt{2}}-\dfrac{3}{\sqrt{2}} i\)
45.
a
\begin{aligned}
& 2,2\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right) \text {, } \\
& 2\left(\cos \frac{4 \pi}{5}+i \sin \frac{4 \pi}{5}\right) \text {, } \\
& 2\left(\cos \frac{6 \pi}{5}+i \sin \frac{6 \pi}{5}\right) \text {, } \\
& 2\left(\cos \frac{8 \pi}{5}+i \sin \frac{8 \pi}{5}\right) \\
\end{aligned}
b \(2, 0.618 + 1.9i, −1.618 + 1.176i, −1.618 − 1.176i, 0.618 − 1.902i\)
47.
a
\begin{aligned}
&4 \left( \cos \dfrac{\pi}{18} + i \sin \dfrac{\pi}{18} \right), \\
&4 \left( \cos \dfrac{13\pi}{18} + i \sin \dfrac{13\pi}{18} \right), \\
&4 \left( \cos \dfrac{25\pi}{18} + i \sin \dfrac{25\pi}{18}\right)
\end{aligned}
b \(1.97 + 0.347i, − 1.286 + 1.532i, − 0.684 − 1.879i\)
49. \(|z|=|\cos \theta+i \sin \theta|=\sqrt{\cos ^2 \theta+\sin ^2 \theta}=1\)
51.
a \(1,\left(\cos \dfrac{2 \pi}{3}+i \sin \dfrac{2 \pi}{3}\right),\left(\cos \dfrac{4 \pi}{3}+i \sin \dfrac{4 \pi}{3}\right)\)
b \(1, i,-1,-i\)
c \(1,\left(\cos \dfrac{2 \pi}{5}+i \sin \dfrac{2 \pi}{5}\right),\left(\cos \dfrac{4 \pi}{5}+i \sin \dfrac{4 \pi}{5}\right),\left(\cos \dfrac{6 \pi}{5}+i \sin \dfrac{6 \pi}{5}\right),\left(\cos \dfrac{8 \pi}{5}+i \sin \dfrac{8 \pi}{5}\right)\)
d \(1,\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right),\left(\cos \dfrac{2 \pi}{3}+i \sin \dfrac{2 \pi}{3}\right),-1,\left(\cos \dfrac{4 \pi}{3}+i \sin \dfrac{4 \pi}{3}\right),\left(\cos \dfrac{5 \pi}{3}+i \sin \dfrac{5 \pi}{3}\right)\)
53. \(\left(\omega_k\right)^n=1^n\left(\cos n \cdot \dfrac{2 \pi k}{n}+i \sin n \cdot \dfrac{2 \pi k}{n}\right)=1(\cos 2 \pi k+i \sin 2 \pi k)=1\)
55.
\begin{aligned}
&8^{1 / 4}\left(\cos \dfrac{3 \pi}{8}+i \sin \dfrac{3 \pi}{8}\right), \\
&8^{1 / 4}\left(\cos \dfrac{5 \pi}{8}+i \sin \dfrac{5 \pi}{8}\right), \\
&8^{1 / 4}\left(\cos \dfrac{11 \pi}{8}+i \sin \dfrac{11 \pi}{8}\right), \\
&8^{1 / 4}\left(\cos \dfrac{13 \pi}{8}+i \sin \dfrac{13 \pi}{8}\right)
\end{aligned}
57.
\begin{aligned}
&\sqrt{2}, \sqrt{2}\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right), \\
&\sqrt{2}\left(\cos \dfrac{2 \pi}{3}+i \sin \dfrac{2 \pi}{3}\right), \quad -\sqrt{2},\\
&\sqrt{2}\left(\cos \dfrac{4 \pi}{3}+i \sin \dfrac{4 \pi}{3}\right) \\
&\sqrt{2}\left(\cos \dfrac{5 \pi}{3}+i \sin \dfrac{5 \pi}{3}\right)
\end{aligned}
59.
\begin{aligned}
& \sqrt{2}\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right) \\
& \sqrt{2}\left(\cos \dfrac{2 \pi}{3}+i \sin \dfrac{2 \pi}{3}\right) \\
& \sqrt{2}\left(\cos \dfrac{4 \pi}{3}+i \sin \dfrac{4 \pi}{3}\right) \\
& \sqrt{2}\left(\cos \dfrac{5 \pi}{3}+i \sin \dfrac{5 \pi}{3}\right)
\end{aligned}
61.
a \(\cos ^2 \theta-\sin ^2 \theta+(2 \sin \theta \cos \theta) i\)
b \(\cos 2 \theta+i \sin 2 \theta\)
c \(\sin 2 \theta=2 \sin \theta \cos \theta ; \quad \cos 2 \theta=\cos ^2 \theta-\sin ^2 \theta\)
63.
a \(\dfrac{b}{a}\)
b \(\dfrac{-a}{b}\)
c \(-1, \dfrac{\pi}{2}\)
65.
a \(z_1 z_2=(a c-b d)+(a d+b c) i\)
b \(a=r \cos \alpha, b=r \sin \alpha, c=R \cos \beta, d=R \sin \beta\)
c \((a c-b d)+(a d+b c) i=(r R \cos \alpha \cos \beta-r R \sin \alpha \sin \beta)+(r R \cos \alpha \sin \beta+r R \sin \alpha \cos \beta) i\)
d \(r R(\cos (\alpha+\beta)+i \sin (\alpha+\beta))\)
10.5 Chapter Summary and Review
Review Problems
1. 
3. 
5. \(\left(\dfrac{-\sqrt{2}}{2}, \dfrac{-\sqrt{2}}{2}\right)\)
7. (0.241, −3.391)
9. \(\left(3 \sqrt{2}, \dfrac{3 \pi}{4}\right)\)
11. \(\left(\sqrt{29}, \tan ^{-1}\left(\dfrac{-2}{5}\right)+\right.2 \pi)\)
13. 
15. 
17. \(x^2+y^2=1\)
19. \(x^2+y^2=(2 x+6)^2\)
21. \(r \cos \theta+r \sin \theta=2\)
23. \(\tan \theta=r\)
25. Circle of radius 3 centered at the origin
27. Circle of radius 3 centered at (3, 0)
29. \(r = 4\)
31. \(r=4 \cos \theta\)
33. \(\left(4, \dfrac{\pi}{6}\right), \quad\left(4, \dfrac{5 \pi}{6}\right)\)
35. \(\left(2 \sqrt{2}, \dfrac{3 \pi}{4}\right)\) and the pole
37. \(4-3 i\)
39. \(-2+4 i\)
41.
a 1
b 1
43.
a -44
b -44
45. \((2 \pm i)^2-4(2 \pm i)+5=(4 \pm 4 i-1)-(8 \pm 4 i)+5=0\)
47. \(z^2+4 z+5\)
49. \(s^2-10 s+41\)
51. 
53.
a \(-1-7 i\)
b \(x^2+2 x+50=0\)
55.
a \(3+\sqrt{2} i\)
b \(x^2-6 x+11=0\)
57. \(5 \sqrt{3}-5 i\)
59. \(5+5 i\)
61. \(3 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)\)
63. \(5(\cos \pi+i \sin \pi)\)
65. \(2\left(\cos \dfrac{4 \pi}{3}+i \sin \dfrac{4 \pi}{3}\right)\)
67.
\begin{aligned}
z_1 z_2 & =16(\cos \pi+i \sin \pi) \\
&= 4\left(\cos \dfrac{-2 \pi}{3}+i \sin \dfrac{-2 \pi}{3}\right) \\
&=-2-2 \sqrt{3} i
\end{aligned}
69.
\begin{aligned}
z_1 z_2 & =\dfrac{5}{2}\left(\cos \dfrac{-\pi}{3}+i \sin \dfrac{-\pi}{3}\right), \\
&=\dfrac{5}{4}-\dfrac{5 \sqrt{3}}{2} i \\
\dfrac{z_1}{z_2} & =10\left(\cos \dfrac{-5 \pi}{6}+i \sin \dfrac{-5 \pi}{6}\right) \\
& =-5 \sqrt{3}-5 i
\end{aligned}
71. 1
73. \(\dfrac{-1}{100}\)
75.
a 
b \(-2\sqrt{2} + 2\sqrt{2}i, 2\sqrt{2} - 2\sqrt{2} i\)
77.
a 
b \(3i, \quad \dfrac{-3\sqrt{3}}{2} - \dfrac{3}{2} i, \quad \dfrac{3\sqrt{3}}{2} - \dfrac{3}{2} i\)
79. \(3 (\cos \theta + i \sin \theta)\), for \(\theta = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{3\pi}{2}, \dfrac{11\pi}{6}\)
81. \(\sqrt{2} (\cos \theta + i \sin \theta )\), for \(\theta = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}\)