Solutions for Polar Form of Complex Numbers

Solutions to Try Its

2. 13 3.

|z|=\sqrt{50}=5\sqrt{2}

z=3\left(\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right)

z=2\left(\cos \left(\frac{\pi }{6}\right)+i\sin \left(\frac{\pi }{6}\right)\right)

z=2\sqrt{3}-2i

{z}_{1}{z}_{2}=-4\sqrt{3};\frac{{z}_{1}}{{z}_{2}}=-\frac{\sqrt{3}}{2}+\frac{3}{2}i

{z}_{0}=2\left(\cos \left(30^\circ \right)+i\sin \left(30^\circ \right)\right)

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

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

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

1. a is the real part, b is the imaginary part, and

i=\sqrt{−1}

3. Polar form converts the real and imaginary part of the complex number in polar form using

x=r\cos\theta

and

y=r\sin\theta

z^{n}=r^{n}\left(\cos\left(n\theta\right)+i\sin\left(n\theta\right)\right)

. It is used to simplify polar form when a number has been raised to a power. 7.

5\sqrt{2}

\sqrt{38}

11.

\sqrt{14.45}

13.

4\sqrt{5}\text{cis}\left(333.4^{\circ}\right)

15.

2\text{cis}\left(\frac{\pi}{6}\right)

17.

\frac{7\sqrt{3}}{2}+i\frac{7}{2}

19.

−2\sqrt{3}−2i

21.

−1.5−i\frac{3\sqrt{3}}{2}

23.

4\sqrt{3}\text{cis}\left(198^{\circ}\right)

25.

\frac{3}{4}\text{cis}\left(180^{\circ}\right)

27.

5\sqrt{3}\text{cis}\left(\frac{17\pi}{24}\right)

29.

7\text{cis}\left(70^{\circ}\right)

31.

5\text{cis}\left(80^{\circ}\right)

33.

5\text{cis}\left(\frac{\pi}{3}\right)

35.

125\text{cis}\left(135^{\circ}\right)

37.

9\text{cis}\left(240^{\circ}\right)

39.

\text{cis}\left(\frac{3\pi}{4}\right)

41.

3\text{cis}\left(80^{\circ}\right)\text{, }3\text{cis}\left(200^{\circ}\right)\text{, }3\text{cis}\left(320^{\circ}\right)

43.

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)

45.

2\sqrt{2}\text{cis}\left(\frac{7\pi}{8}\right)\text{, }2\sqrt{2}\text{cis}\left(\frac{15\pi}{8}\right)

47.

49.

51.

53.

55.

57.

3.61e^{−0.59i}

59.

−2+3.46i

61.

−4.33−2.50i

Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.