Expressing complex numbers in standard form problem

Question

Express the following complex number in the standard form $x + iy$

$ie^{\frac{i\pi}{2} +3} $

I have made an attempt and got the answer $\cos(\frac{\pi}{2} +3) +i\sin(\frac{\pi}{2} +3)$. Is this an acceptable result? This is a non-calculator question.

Answer 1

Here's what I did:

I used the identity $e^{ix}=\sin(x)+i\cos(x)$

I then split the probelm apart into this form: $$ie^{ \frac{i\pi}{2}}e^3$$ Then I applied the identity above: $$ie^3 \left( \cos \left( \frac{\pi}{2} \right) + i \sin \left( \frac{\pi}{2} \right) \right) $$ Evaluating the trig functions, I get: $$i^2e^3$$ And note that $i^2=-1$, so the final form is: $$-e^3$$

Note: if your teacher really wants it in the form $x+iy$, you can make it $-e^3+0i$

Answer 2

So.. using Eulers identity for complex numbers, we know that for $z=a+ib$:

$e^z=e^a(\text{cos}(b)+i\text{sin}(b))$

So your number becomes:

$ie^{\frac{i\pi}{2}+3}=ie^3\left(\text{cos}(\frac{\pi}{2})+i\text{sin}(\frac{\pi}{2})\right)=e^3\left(i\text{cos}(\frac{\pi}{2})-\text{sin}(\frac{\pi}{2})\right)$