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?
$\endgroup$ 22 Answers
$\begingroup$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 $\sqrt 8 (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4} ) = (2\sqrt 2)(\frac 1{\sqrt 2})(1+i) = 2(1+i) = 2 + 2i$
$\endgroup$ 2 $\begingroup$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
$$ \sqrt{8}\left(\cos \frac{\pi}{4} + i \sin\frac{\pi}{4}\right) = \sqrt{8}e^{i\pi/4} $$
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