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

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Answer 1 · 2017-08-20T03:36:34

Use Euler's identity...

...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:

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

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

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