Exercises: 10.4 Polar Form of Complex Numbers

1.

$1,i,i^2,i^3$ and $i^4$

2.

$-1,-i,-i^2,-i^3$ and $-i^4$

3.

$1+2i$ and $i(1+2i)$

4.

$3-4i$ and $i(3-4i)$

5.

$6(\cos {\displaystyle \frac{2\pi }{3}}+i\sin {\displaystyle \frac{2\pi }{3}})$

6.

$4(\cos {\displaystyle \frac{7\pi }{4}}+i\sin {\displaystyle \frac{7\pi }{4}})$

7.

$\sqrt{2}(\cos {\displaystyle \frac{3\pi }{4}}+i\sin {\displaystyle \frac{3\pi }{4}})$

8.

${\displaystyle \frac{3}{2}}(\cos {\displaystyle \frac{5\pi }{6}}+i\sin {\displaystyle \frac{5\pi }{6}})$

9.

$5(\cos 5.2+i\sin 5.2)$

10.

$3(\cos 3.5+i\sin 3.5)$

11.

$12(\cos 115°+i\sin 115°)$

12.

$20(\cos 250°+i\sin 250°)$

13.

$3i$ and $-3i$

14.

$2+2i$ and $2-2i$

15.

$-3-\sqrt{3}i$ and $3-\sqrt{3}i$

16.

$2\sqrt{3}+2i$ and $-2\sqrt{3}+2i$

17.

$-4+2i$ and $4-2i$

18.

$-3-8i$ and $3+8i$

19.

$9-5i$ and $9+5i$

20.

$2+6i$ and $2-6i$

21.

$3+4i,3-4i,-3+4i,$ and $-3-4i$

22.

$1+3i,1-3i,-1+3i,$ and $-1-3i$

23.

What can you conclude about the polar forms of $z$ and $\overline{z}\text{?}$

24.

What can you conclude about the polar forms of $z$ and $-z\text{?}$

25.

$z_1=4(\cos {\displaystyle \frac{4\pi }{3}}+i\sin {\displaystyle \frac{4\pi }{3}})$ $z_2={\displaystyle \frac{1}{2}}(\cos {\displaystyle \frac{5\pi }{6}}+i\sin {\displaystyle \frac{5\pi }{6}})$

26.

$z_1=6(\cos {\displaystyle \frac{5\pi }{8}}+i\sin {\displaystyle \frac{5\pi }{8}})$ $z_2={\displaystyle \frac{3}{2}}(\cos {\displaystyle \frac{\pi }{8}}+i\sin {\displaystyle \frac{\pi }{8}})$

27.

$z_1=3(\cos {\displaystyle \frac{3\pi }{5}}+i\sin {\displaystyle \frac{3\pi }{5}})$ $z_2=2(\cos {\displaystyle \frac{3\pi }{10}}+i\sin {\displaystyle \frac{3\pi }{10}})$

28.

$z_1=4(\cos {\displaystyle \frac{5\pi }{12}}+i\sin {\displaystyle \frac{5\pi }{12}})$ $z_2=6(\cos {\displaystyle \frac{3\pi }{4}}+i\sin {\displaystyle \frac{3\pi }{4}})$

29.

$z_1=2i,z_2=4i$

30.

$z_1=-2i,z_2=3i$

31.

$z_1=2\sqrt{3}-2i\text{,}$ $z_2=-1+i$

32.

$z_1=\sqrt{3}+i\text{,}$ $z_2=-1+\sqrt{3}$

33.

$(2+2i{)}^5$

34.

$(\sqrt{2}-\sqrt{2}i{)}^6$

35.

$(-1+\sqrt{3}i{)}^8$

36.

$({\displaystyle \frac{1}{2}}+{\displaystyle \frac{\sqrt{3}}{2}}i{)}^{12}$

37.

$(\sqrt{3}-i{)}^{10}$

38.

$(1-i{)}^{20}$

39.

$2-2i$

40.

$3+\sqrt{3}i$

41.

$-\sqrt{2}+\sqrt{6}i$

42.

$-1-i$

43.

The square roots of $9i\text{.}$

44.

The fourth roots of $-81\text{.}$

45.

The fifth roots of $32\text{.}$

46.

The cube roots of $i\text{.}$

47.

The cube roots of $4\sqrt{3}+4i\text{.}$

48.

The square roots of $-2+2\sqrt{3}i\text{.}$

49.

Show that any complex number of the form $z=\cos \theta +i\sin \theta$ lies on the unit circle in the complex plane.

50.

Show that if $|z|=1\text{,}$ then ${\displaystyle \frac{1}{z}}=\overline{z}\text{.}$

51.

1. Find three distinct cube roots of 1.
2. Find four distinct fourth roots of 1.
3. Find five distinct fifth roots of 1.
4. Find six distinct sixth roots of 1.

52.

1. Find the sum of the three distinct cube roots of 1. (Hint: Plot the roots.)
2. Find the sum of the four distinct fourth roots of 1.
3. Find the sum of the five distinct fifth roots of 1.
4. Find the sum of the six distinct sixth roots of 1.

53.

If $n$ is a positive integer, define

${\omega }_k=\cos {\displaystyle \frac{2\pi k}{n}}+i\sin {\displaystyle \frac{2\pi k}{n}}$

for $k=0,1,2,\cdots ,n-1.$ Show that $({\omega }_k{)}^n=1\text{.}$ (We call ${\omega }_k$ an $n^{th}$ root of unity.)

54.

Let $\omega =\cos {\displaystyle \frac{2\pi }{n}}+i\sin {\displaystyle \frac{2\pi }{n}}\text{,}$ where $n$ is a positive integer. Show that the $n$ distinct $n^{th}$ roots of unity are $\omega ,{\omega }^2,{\omega }^3,\cdots ,{\omega }^{n-1}\text{.}$

55.

$z^4+4z^2+8=0$

56.

$z^6+4z^3+8=0$

57.

$z^6-8=0$

58.

$z^4-9i=0$

59.

$z^4+2z^2+4=0$

60.

$z^4-2z^2+4=0$

61.

1. Let $z=\cos \theta +i\sin \theta \text{.}$ Compute $z^2$ by expanding the product.
2. Use DeMoivre’s theorem to compute $z^2\text{.}$
3. Compare your answers to (a) and (b) to write identities for $\sin 2\theta$ and $\cos 2\theta \text{.}$

62.

1. Let $z=\cos \theta +i\sin \theta \text{.}$ Compute $z^3$ by expanding the product.
2. Use De Moivre’s theorem to compute $z^3\text{.}$
3. Compare your answers to (a) and (b) to write identities for $\sin 3\theta$ and $\cos 3\theta \text{.}$

63.

Suppose that $z=a+bi$ and that the real numbers $a$ and $b$ are both nonzero.

1. What is the slope of the segment in the complex plane joining the origin to $z\text{?}$
2. What is the slope of the segment in the complex plane joining the origin to $zi\text{?}$
3. What is the product of the slopes of the two segments from parts (a) and (b)? What can you conclude about the angle between the two segments?

64.

Suppose that $z=a+bi$ and that $a$ and $b$ are both real numbers.

1. If $a\ne 0$ and $b=0\text{,}$ then what is the slope of the segment in the complex plane joining the origin to $z\text{?}$ What is the slope of the segment joining the origin to $zi\text{?}$
2. If $a=0$ and $b\ne 0\text{,}$ then what is the slope of the segment in the complex plane joining the origin to $z\text{?}$ What is the slope of the segment joining the origin to $zi\text{?}$
3. What can you conclude about the angle between the two segments from parts (a) and (b)?

65.

Prove the product rule by following the steps.

1. Suppose $z_1=a+bi$ and $z_2=c+di\text{.}$ Compute $z_1{z}_2\text{.}$
2. Now suppose that $z_1=r(\cos \alpha +i\sin \alpha )$ and $z_2=R(\cos \beta +i\sin \beta )\text{.}$ Write $a,b,c$ and $d$ in terms of $r,R,\alpha$ and $\beta \text{.}$
3. Substitute your expressions for $a,b,c$ and $d$ into your formula for $z_1{z}_2\text{.}$
4. Use the laws of sines and cosines to simplify your answer to part (c).

66.

Let $z_1=r(\cos \alpha +i\sin \alpha )$ and $z_2=R(\cos \beta +i\sin \beta )\text{.}$ Prove the quotient rule as follows: Set $w={\displaystyle \frac{r}{R}}(\cos (\alpha -\beta )+i\sin (\alpha -\beta )),$ and show that $z_1=w{z}_2\text{.}$