How to find real and imaginary part of $i^{-n}$

Question

How do I find real and imaginary part of

$$ i^{-n}, \ n \in \mathbb{N}. $$

Attempt:

Inserting n = 1, 2, 3, 4 ...

Trying to find a pattern.

n = 1 $$ 1/i^1 = i^{-1} = -i $$

n = 2

$$ 1/i^{2} = -1 $$

n = 3

$$ 1/i^{3} = i $$

n = 4

$$ 1/i^{4} = 1 $$

I see some sort of a trig function pattern but I don't have any idea on how to put it.

Answer 1

Hint: $$i = e^{i(\pi/2 + 2k\pi)},$$ where $k$ is an integer.

Answer 2

Hint: $\displaystyle i^{-n} = \frac{1}{i^n} = \left(\frac{1}{i}\right)^n, \quad \frac{1}{i} = -i$