# Convert complex to polar cord

Write the following numbers in the polar form $$๐๐^{๐๐}, โ๐<๐โค๐$$:

$$r$$ = ?, $$\theta$$ =?

$$-\dfrac{\sqrt{7}(1+i)}{\sqrt{2}+ i}$$

When i used wolfram I got $$r = 2.16025$$ and $$\theta = -170.264^{ยฐ}$$. However, when I input this in WebAssign I got the right answer for $$r$$ but the wrong answer for $$\theta$$. I then tried doing $$tan(-170.264^{ยฐ}) =.171579$$ (in degree) and $$tan(-170.264) = -.710889017$$ (in radians), and they were both wrong. I'm lost what should be $$\theta$$ ?

• why would you take tan(-170.265)? Nov 13, 2019 at 8:10
• Put your software down and draw some pictures. If you sketch $-\sqrt{7} (1+i)$ you can see it makes an angle of 235 degrees anticlockwise from the positive real axis. Similarly, $\sqrt{2} + i$ in the first quadrant makes an angle of 35.26 degrees. If you divide two complex numbers you subtract their arguments which is 189.74 degrees or -170.26 degrees if you measure clockwise from the positive real axis.
– Paul
Nov 13, 2019 at 8:45
• Don't use degrees.
– user65203
Nov 13, 2019 at 9:40

$$\frac{\sqrt7\sqrt2}{\sqrt3}$$
$$-\pi+\frac{\pi}4-\arctan\frac1{\sqrt2}.$$
($$-\pi$$ is chosen rather than $$\pi$$ so that the sum be in the allowed range.)
Firstly, you can rewrite $$\frac{โ\sqrt7(1+i)}{\sqrt2+i}$$ as $$-\frac{\sqrt7+\sqrt14}{3} + i(\frac{\sqrt7 - \sqrt14}{3})$$. Then use that $$re^{i\theta} = r(cos(\theta) + isin(\theta))$$ and you get that $$\begin{equation*} \begin{cases} r\cos(\theta) = -\frac{\sqrt7+\sqrt14}{3}\\ r\sin(\theta) = \frac{\sqrt7 - \sqrt14}{3} \end{cases} \end{equation*}$$ $$or$$ $$\begin{equation*} \begin{cases} \tan(\theta) = -\frac{\sqrt7-\sqrt14}{\sqrt7+\sqrt14}\\ r^2 = \frac{14}{3} \end{cases} \end{equation*}$$