Converting Complex numbers into Cartesian Form

Question

I am trying to convert the following Complex number equation into Cartesian Form:

$$ \sqrt{8}\left(\cos \frac{\pi}{4} + i \sin\frac{\pi}{4}\right) $$

So far I have tried doing both:

$\sqrt{8} \frac{\pi}{4}\cos \frac{\pi}{4}$ and $\sqrt{8}\frac{\pi}{4}\sin \frac{\pi}{4}$ which returns $(\frac{\pi}{2}, \frac{\pi}{2})$ but the tutorial I am following says this is the incorrect answer.. Does anyone know how to correctly convert Complex numbers in Polar form to Cartesian Form?

Answer 1

A direct relation between the cartesian and polar representation a complex number is provided by Euler's formula

$$ re^{i\theta} = r\cos\theta + i\sin\theta $$

You should compare this agains the number you have, which allows to conclude that

$$ r = \sqrt{8} ~~~\mbox{and}~~~ \theta = \pi/4 $$

that is

$$ \sqrt{8}\left(\cos \frac{\pi}{4} + i \sin\frac{\pi}{4}\right) = \sqrt{8}e^{i\pi/4} $$

Answer 2

It's as simple as multiplying it out.

$\sin \frac{\pi}{4} =\cos \frac{\pi}{4} = \frac 1{\sqrt 2}$ so $\cos \frac{\pi}{4} + i \sin \frac{\pi}{4} = \frac 1{\sqrt 2}(1+i)$

So $\sqrt 8 (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4} ) = (2\sqrt 2)(\frac 1{\sqrt 2})(1+i) = 2(1+i) = 2 + 2i$