- #1

MatinSAR

- 565

- 179

- Homework Statement
- Stated below.

- Relevant Equations
- Complex algebra.

**Question 1:**Find the modulus and argument of ##z=-\sin \frac {\pi}{8}-i\cos \frac {\pi}{8}##.

The modulus is obviously 1. I can't prove that the argument is ##\frac {-5\pi} {8}##. I think ##\frac {-5\pi} {8}## is not correct ...

What I've done:

$$\tan \theta=\cot \frac {\pi}{8}$$$$\tan \theta=\dfrac {1+\cos \frac {\pi}{4}}{\sin \frac {\pi}{4}}$$$$\tan \theta=1+ \sqrt 2$$$$\theta=\arctan (1+ \sqrt 2) +2k\pi$$

I can't find out why it should be equal to ##\frac {-5\pi} {8}##.

**Question 2:**Calculate ##(-1+i\sqrt 3)^{60} ##.

We first write it in polar form ##re^{i\theta}##.

$$r=2$$$$\tan \theta = -\sqrt 3 $$$$ \theta = \dfrac {2\pi}{3}$$

We have:

$$ (2e^{i(\dfrac {2\pi}{3})})^{60} =2^{60}e^{i(40\pi)}=2^{60}(\cos 40\pi + i\sin 40\pi)=2^{60}$$

Am I right?

Many thanks.