Chapter 10: Polar Coordinates and Complex Numbers

Exercises: 10.4 Polar Form of Complex Numbers

SKILLS

Practice each skill in the Homework Problems listed.

  1. Convert from polar form to standard form #5–12
  2. Write a complex number in polar form #13-22
  3. Find the product or quotient of two complex numbers in polar form #25–32
  4. Find a power of a complex number #33–42
  5. Find the complex roots of a number #43–48, 51–52, 55–60

 

Suggested Homework

Problems: #6, 12, 14, 20, 28, 36, 42, 44, 58

 

Exercises for 10.4 Polar Form of Complex Numbers

Exercise Group

For Problems 1–4, simplify and plot each complex number as a point on the complex plane.

1.

[latex]1,~i,~i^2,~i^3[/latex] and [latex]i^4[/latex]

2.

[latex]-1,~-i,~-i^2,~-i^3[/latex] and [latex]-i^4[/latex]

3.

[latex]1+2i[/latex] and [latex]i(1+2i)[/latex]

4.

[latex]3-4i[/latex] and [latex]i(3-4i)[/latex]

Exercise Group

For Problems 5–8, write the complex numbers in standard form. Give exact values for your answers.

5.

[latex]6\left(\cos \dfrac{2\pi}{3} + i\sin \dfrac{2\pi}{3}\right)[/latex]

6.

[latex]4\left(\cos \dfrac{7\pi}{4} + i\sin \dfrac{7\pi}{4}\right)[/latex]

7.

[latex]\sqrt{2}\left(\cos \dfrac{3\pi}{4} + i\sin \dfrac{3\pi}{4}\right)[/latex]

8.

[latex]\dfrac{3}{2}\left(\cos \dfrac{5\pi}{6} + i\sin \dfrac{5\pi}{6}\right)[/latex]

Exercise Group

For Problems 9–12, write the complex numbers in standard form. Round your answers to hundredths.

9.

[latex]5(\cos 5.2 + i\sin 5.2)[/latex]

10.

[latex]3(\cos 3.5 + i\sin 3.5)[/latex]

11.

[latex]12(\cos 115° + i\sin 115°)[/latex]

12.

[latex]20(\cos 250° + i\sin 250°)[/latex]

Exercise Group

For Problems 13–16, write the complex numbers in polar form. Give exact values for your answers.

13.

[latex]3i[/latex] and [latex]-3i[/latex]

14.

[latex]2+2i[/latex] and [latex]2-2i[/latex]

15.

[latex]-3-\sqrt{3}i[/latex] and [latex]3-\sqrt{3}i[/latex]

16.

[latex]2\sqrt{3}+2i[/latex] and [latex]-2\sqrt{3}+2i[/latex]

Exercise Group

For Problems 17–22, write the complex numbers in polar form. Round your answers to hundredths.

17.

[latex]-4+2i[/latex] and [latex]4-2i[/latex]

18.

[latex]-3-8i[/latex] and [latex]3+8i[/latex]

19.

[latex]9-5i[/latex] and [latex]9+5i[/latex]

20.

[latex]2+6i[/latex] and [latex]2-6i[/latex]

21.

[latex]3+4i,~ 3-4i,~ -3+4i,[/latex] and [latex]-3-4i[/latex]

22.

[latex]1+3i,~ 1-3i,~ -1+3i,[/latex] and [latex]-1-3i[/latex]

23.

What can you conclude about the polar forms of [latex]z[/latex] and [latex]\bar{z}\text{?}[/latex]

24.

What can you conclude about the polar forms of [latex]z[/latex] and [latex]-z\text{?}[/latex]

Exercise Group

For Problems 25–28, find the product [latex]z_1z_2[/latex] and the quotient [latex]\dfrac{z_1}{z_2}\text{.}[/latex]

25.

[latex]z_1=4\left(\cos \dfrac{4\pi}{3} + i\sin \dfrac{4\pi}{3}\right)[/latex] [latex]z_2=\dfrac{1}{2}\left(\cos \dfrac{5\pi}{6} + i\sin \dfrac{5\pi}{6}\right)[/latex]

26.

[latex]z_1=6\left(\cos \dfrac{5\pi}{8} + i\sin \dfrac{5\pi}{8}\right)[/latex] [latex]z_2=\dfrac{3}{2}\left(\cos \dfrac{\pi}{8} + i\sin \dfrac{\pi}{8}\right)[/latex]

27.

[latex]z_1=3\left(\cos \dfrac{3\pi}{5} + i\sin \dfrac{3\pi}{5}\right)[/latex] [latex]z_2=2\left(\cos \dfrac{3\pi}{10} + i\sin \dfrac{3\pi}{10}\right)[/latex]

28.

[latex]z_1=4\left(\cos \dfrac{5\pi}{12} + i\sin \dfrac{5\pi}{12}\right)[/latex] [latex]z_2=6\left(\cos \dfrac{3\pi}{4} + i\sin \dfrac{3\pi}{4}\right)[/latex]

Exercise Group

For Problems 29–32, convert the complex number to polar form, then find the product [latex]z_1z_2[/latex] and the quotient [latex]\dfrac{z_1}{z_2}\text{.}[/latex]

29.

[latex]z_1=2i,~z_2=4i[/latex]

30.

[latex]z_1=-2i,~z_2=3i[/latex]

31.

[latex]z_1=2\sqrt{3}-2i\text{,}[/latex] [latex]~z_2=-1+i[/latex]

32.

[latex]z_1=\sqrt{3}+i\text{,}[/latex] [latex]~z_2=-1+\sqrt{3}[/latex]

Exercise Group

For Problems 33–38, find the power.

33.

[latex](2+2i)^5[/latex]

34.

[latex](\sqrt{2}-\sqrt{2}i)^6[/latex]

35.

[latex](-1+\sqrt{3}i)^8[/latex]

36.

[latex](\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)^{12}[/latex]

37.

[latex](\sqrt{3}-i)^{10}[/latex]

38.

[latex](1-i)^{20}[/latex]

Exercise Group

For Problems 39–42, use De Moivre’s theorem to find the reciprocal.

39.

[latex]2-2i[/latex]

40.

[latex]3+\sqrt{3}i[/latex]

41.

[latex]-\sqrt{2}+\sqrt{6}i[/latex]

42.

[latex]-1-i[/latex]

Exercise Group

For Problems 43–48,

  1. Find the roots and plot them in the complex plane.
  2. Write the roots in standard form.
43.

The square roots of [latex]9i\text{.}[/latex]

44.

The fourth roots of [latex]-81\text{.}[/latex]

45.

The fifth roots of [latex]32\text{.}[/latex]

46.

The cube roots of [latex]i\text{.}[/latex]

47.

The cube roots of [latex]4\sqrt{3}+4i\text{.}[/latex]

48.

The square roots of [latex]-2+2\sqrt{3}i\text{.}[/latex]

49.

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

50.

Show that if [latex]{|z|}=1\text{,}[/latex] then [latex]\dfrac{1}{z}=\bar{z}\text{.}[/latex]

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 [latex]n[/latex] is a positive integer, define

[latex]\omega_k = \cos \dfrac{2\pi k}{n} + i\sin \dfrac{2\pi k}{n}[/latex]

for [latex]k = 0,~1,~2,~ \cdots,~ n-1.[/latex] Show that [latex](\omega_k)^n = 1\text{.}[/latex] (We call [latex]\omega_k[/latex] an [latex]n^{th}[/latex] root of unity.)

54.

Let [latex]\omega = \cos \dfrac{2\pi}{n} + i\sin \dfrac{2\pi}{n}\text{,}[/latex] where [latex]n[/latex] is a positive integer. Show that the [latex]n[/latex] distinct [latex]n^{th}[/latex] roots of unity are [latex]\omega,~\omega^2,~\omega^3,~ \cdots,~\omega^{n-1}\text{.}[/latex]

Exercise Group

For Problems 55-60, solve the equation.

55.

[latex]z^4+4z^2+8=0[/latex]

56.

[latex]z^6+4z^3+8=0[/latex]

57.

[latex]z^6-8=0[/latex]

58.

[latex]z^4-9i=0[/latex]

59.

[latex]z^4+2z^2+4=0[/latex]

60.

[latex]z^4-2z^2+4=0[/latex]

61.

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

62.

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

Exercise Group

Problems 63 and 64 show that multiplication by [latex]i[/latex] results in a rotation of [latex]90°\text{.}[/latex]

63.

Suppose that [latex]z=a+bi[/latex] and that the real numbers [latex]a[/latex] and [latex]b[/latex] are both nonzero.

  1. What is the slope of the segment in the complex plane joining the origin to [latex]z\text{?}[/latex]
  2. What is the slope of the segment in the complex plane joining the origin to [latex]zi\text{?}[/latex]
  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 [latex]z=a+bi[/latex] and that [latex]a[/latex] and [latex]b[/latex] are both real numbers.

  1. If [latex]a \not= 0[/latex] and [latex]b=0\text{,}[/latex] then what is the slope of the segment in the complex plane joining the origin to [latex]z\text{?}[/latex] What is the slope of the segment joining the origin to [latex]zi\text{?}[/latex]
  2. If [latex]a = 0[/latex] and [latex]b \not=0\text{,}[/latex] then what is the slope of the segment in the complex plane joining the origin to [latex]z\text{?}[/latex] What is the slope of the segment joining the origin to [latex]zi\text{?}[/latex]
  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 [latex]z_1=a+bi[/latex] and [latex]z_2=c+di\text{.}[/latex] Compute [latex]z_1z_2\text{.}[/latex]
  2. Now suppose that [latex]z_1=r(\cos \alpha +i \sin \alpha)[/latex] and [latex]z_2=R(\cos \beta +i \sin \beta)\text{.}[/latex] Write [latex]a,~b,~c[/latex] and [latex]d[/latex] in terms of [latex]r,~R,~\alpha[/latex] and [latex]\beta\text{.}[/latex]
  3. Substitute your expressions for [latex]a,~b,~c[/latex] and [latex]d[/latex] into your formula for [latex]z_1z_2\text{.}[/latex]
  4. Use the laws of sines and cosines to simplify your answer to part (c).

66.

Let [latex]z_1=r(\cos \alpha +i \sin \alpha)[/latex] and [latex]z_2=R(\cos \beta +i \sin \beta)\text{.}[/latex] Prove the quotient rule as follows: Set [latex]w=\dfrac{r}{R}(\cos (\alpha - \beta) + i\sin (\alpha - \beta)),[/latex] and show that [latex]z_1=wz_2\text{.}[/latex]

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