Solutions for Polar Form of Complex Numbers

Solutions to Try Its

1.
Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).

2. 13

3. [latex]|z|=\sqrt{50}=5\sqrt{2}[/latex]

4. [latex]z=3\left(\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right)[/latex]

5. [latex]z=2\left(\cos \left(\frac{\pi }{6}\right)+i\sin \left(\frac{\pi }{6}\right)\right)[/latex]

6. [latex]z=2\sqrt{3}-2i[/latex]

7. [latex]{z}_{1}{z}_{2}=-4\sqrt{3};\frac{{z}_{1}}{{z}_{2}}=-\frac{\sqrt{3}}{2}+\frac{3}{2}i[/latex]

8. [latex]{z}_{0}=2\left(\cos \left(30^\circ \right)+i\sin \left(30^\circ \right)\right)[/latex]

[latex]{z}_{1}=2\left(\cos \left(120^\circ \right)+i\sin \left(120^\circ \right)\right)[/latex]

[latex]{z}_{2}=2\left(\cos \left(210^\circ \right)+i\sin \left(210^\circ \right)\right)[/latex]

[latex]{z}_{3}=2\left(\cos \left(300^\circ \right)+i\sin \left(300^\circ \right)\right)[/latex]

Solutions to Odd-Numbered Answers

1. a is the real part, b is the imaginary part, and [latex]i=\sqrt{−1}[/latex]

3. Polar form converts the real and imaginary part of the complex number in polar form using [latex]x=r\cos\theta[/latex] and [latex]y=r\sin\theta[/latex]

5. [latex]z^{n}=r^{n}\left(\cos\left(n\theta\right)+i\sin\left(n\theta\right)\right)[/latex]. It is used to simplify polar form when a number has been raised to a power.

7. [latex]5\sqrt{2}[/latex]

9. [latex]\sqrt{38}[/latex]

11. [latex]\sqrt{14.45}[/latex]

13. [latex]4\sqrt{5}\text{cis}\left(333.4^{\circ}\right)[/latex]

15. [latex]2\text{cis}\left(\frac{\pi}{6}\right)[/latex]

17. [latex]\frac{7\sqrt{3}}{2}+i\frac{7}{2}[/latex]

19. [latex]−2\sqrt{3}−2i[/latex]

21. [latex]−1.5−i\frac{3\sqrt{3}}{2}[/latex]

23. [latex]4\sqrt{3}\text{cis}\left(198^{\circ}\right)[/latex]

25. [latex]\frac{3}{4}\text{cis}\left(180^{\circ}\right)[/latex]

27. [latex]5\sqrt{3}\text{cis}\left(\frac{17\pi}{24}\right)[/latex]

29. [latex]7\text{cis}\left(70^{\circ}\right)[/latex]

31. [latex]5\text{cis}\left(80^{\circ}\right)[/latex]

33. [latex]5\text{cis}\left(\frac{\pi}{3}\right)[/latex]

35. [latex]125\text{cis}\left(135^{\circ}\right)[/latex]

37. [latex]9\text{cis}\left(240^{\circ}\right)[/latex]

39. [latex]\text{cis}\left(\frac{3\pi}{4}\right)[/latex]

41. [latex]3\text{cis}\left(80^{\circ}\right)\text{, }3\text{cis}\left(200^{\circ}\right)\text{, }3\text{cis}\left(320^{\circ}\right)[/latex]

43. [latex]2\sqrt[3]{4}\text{cis}\left(\frac{2\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{8\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{14\pi}{9}\right)[/latex]

45. [latex]2\sqrt{2}\text{cis}\left(\frac{7\pi}{8}\right)\text{, }2\sqrt{2}\text{cis}\left(\frac{15\pi}{8}\right)[/latex]

47.
Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).

49.
Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).

51.
Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).

53.
Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).

55.
Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).

57. [latex]3.61e^{−0.59i}[/latex]

59. [latex]−2+3.46i[/latex]

61. [latex]−4.33−2.50i[/latex]