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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:

$\sqrt{8} (\pi/4)\cos (\pi/4)$ and $\sqrt{8}(\pi/4)\sin (\pi/4)$ which returns $(0.5\pi, 0.5\pi)$ 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?

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  • $\begingroup$ How did you go from $\sqrt8\cos\frac\pi4$ to $\sqrt8\frac\pi4\cos\frac\pi4$? $\endgroup$ May 14, 2017 at 13:06
  • $\begingroup$ You may not like it, but this is the Cartesian form (i.e., $z=x+iy$). The polar form (as suggested below) is $z=re^{i\theta}=\sqrt{8}e^{i\pi/4}$. $\endgroup$ Jul 30, 2017 at 20:45

2 Answers 2

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

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  • $\begingroup$ So simple, thanks for pointing it out. If the equation was changed to $(cos - pi/4 + i sin - pi/4)$ would there be a different method for that or simply calculate $(cos(-pi/4)$ and then use the method above? $\endgroup$
    – bancqm
    May 14, 2017 at 14:16
  • $\begingroup$ Just work out the respective trig ratios the same way. So that would be $2-2i$ $\endgroup$
    – Deepak
    May 14, 2017 at 23:26

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