Question

(4(cos135 + i sin 135))^3

Answer 1

4(exp(i*135))^3

Answer 2

is that in degrees or radians?

Answer 3

\[(4\cos135+(i)\sin135)^{3}\] is this the correct equation? (I'm assuming its 135 degrees)
If you look at the unit circle you will notice that 135 degrees is the same as \[ \frac{ 3\pi }{ 4 }\]
sin of \[ \frac{ 3\pi }{ 4 }\] is the same as \[\frac{ \sqrt{2} }{ 2 }\] and cos of \[ \frac{ 3\pi }{ 4 }\] is \[\frac{ -\sqrt{2} }{ 2 }\]
so that simplifies things a little to: \[(4(\frac{ -\sqrt{2} }{ 2 }) + (i)(\frac{ \sqrt{2} }{ 2 }))^3\]
then lets split this up to:
\[4(\frac{ -\sqrt{2} }{ 2 }) = -(i)(\frac{ \sqrt{2} }{ 2 })\]
Then divide both sides by \[\frac{ \sqrt{2} }{ 2 }\]

and you get \[i = -4\]

Answer 4

@natecmcgregor \(i\) isn't a variable here; \(i=\sqrt{-1}\)

Answer 5

Well its not very clear as to what anything is, no instructions were posted.

Answer 6

Instead, use DeMoivre's theorem:
\[\large (r(\cos\theta+i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\]