Question

How can you use trigonometric functions to simplify `15 e^( ( 3 pi)/8 i )` into a non-exponential complex number?

Answer 1

`15e^((3pi)/8i) = 15/2(sqrt(2-sqrt(2)) + sqrt(2+sqrt(2))i)`

Explanation:

Using the identity `e^(itheta) = cos(theta) + isin(theta)` we have:

`15e^((3pi)/8i) = 15(cos((3pi)/8)+isin((3pi)/8))`

If we do not want trig functions either, we can simplify further using the half angle formulas.

`cos((3pi)/8) = cos(1/2((3pi)/4)) = sqrt((1+cos((3pi)/4))/2) = sqrt(2-sqrt(2))/2`

`sin((3pi)/8) = sin(1/2((3pi)/4)) = sqrt((1-cos((3pi)/4))/2) = sqrt(2+sqrt(2))/2`

Thus

`15e^((3pi)/8i) = 15(sqrt(2-sqrt(2))/2 + sqrt(2+sqrt(2))/2i)`

`= 15/2(sqrt(2-sqrt(2)) + sqrt(2+sqrt(2))i)`