Finding all possible solutons

Question

I'm working through "Calculus: Single Variable Part 1 - Functions" on Coursera platform and asked :

Find all possible solutions to the equation : $e^{ix} = i$

How to I begin with such a question ? I need to find the range of function of $e^{\sqrt-1x} = \sqrt-1$ , this means to find the range of values of x ?

I've researched these questions :

Finding all possible values

Finding all possible values of a Function

My conclusion is that while "Finding all possible solutions" is a common question the solutions to such type questions are unique.

Answer 1

$$e^{ix}= \cos x+ i \sin x $$

$$e^{ix}=i$$

So,$$\cos x+ i \sin x=i $$

$$\cos x=i(1-\sin x) $$

This is true only when both $\cos x=0$ and $\sin x=1$.

$x=\frac{\pi}{2},\frac{5\pi}{2},\frac{9\pi}{2},\frac{-3\pi}{2},\frac{-7\pi}{2}, ...$

So, $x= (4n+1)\frac{\pi}{2}$ for integer $n$.

Answer 2

We know that : $$e^{i\theta} = \cos \theta + i\sin \theta $$

Now, $$e^{i\frac {\pi}{2}}=? $$ Hope you can take it from here.

Answer 3

Think geometrically. $e^{ix}$ represents the circle of radius 1 around the origin (every complex number can be written in polar form as $re^{i\theta}$ where $r$ is the modulus and $\theta$ is the angle with 0 in the positive real axis direction).

With this knowledge, what angle(s) $x$ will point you due north, where $i$ is?

Note: This is saying much the same thing as Euler's formula, since representing the exponential with sines and cosines is giving a polar representation.