Question

How can you use trigonometric functions to simplify `4 e^( ( 5 pi)/4 i )` into a non-exponential complex number?

Answer 1

Use the Moivre formula.

Explanation:

The Moivre formula tells us that `e^(itheta) = cos(theta) + isin(theta)`.

Apply this here : `4e^(i(5pi)/4) = 4(cos((5pi)/4) + isin((5pi)/4))`

On the trigonometric circle, `(5pi)/4 = (-3pi)/4`. Knowing that `cos((-3pi)/4) = -sqrt2/2` and `sin((-3pi)/4) = -sqrt2/2`, we can say that `4e^(i(5pi)/4) = 4(-sqrt2/2 -i(sqrt2)/2) = -2sqrt2 -2isqrt2`.