How do you evaluate e^( ( 7 pi)/4 i) - e^( ( 2 pi)/3 i) using trigonometric functions?

1 Answer
Aug 20, 2017

Use Euler's identity...

Explanation:

...which I will use, but not prove, here:

e^(ix) = cosx + isinx

so, for the first term, we have x = (7pi)/4
so the first term can be re-written:

cos((7pi)/4) + isin((7pi)/4)

and the second can be rewritten:

cos((2pi)/3) + isin((2pi)/3)

So, plugging it all back in:

(cos((7pi)/4) + isin((7pi)/4)) - (cos((2pi)/3) + isin((2pi)/3))

= (1/sqrt(2) - 1/sqrt(2)i) - (-1/sqrt(2) + sqrt(3)/2i)