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

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

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Answer 1 · 2017-08-20T03:36:34

Use Euler's identity...

Explanation:

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

$e^{i x} = cos x + i sin x$ $e^(ix) = cosx + isinx$

so, for the first term, we have $x = \frac{7 π}{4}$ $x = (7pi)/4$
so the first term can be re-written:

$cos (\frac{7 π}{4}) + i sin (\frac{7 π}{4})$ $cos((7pi)/4) + isin((7pi)/4)$

and the second can be rewritten:

$cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3})$ $cos((2pi)/3) + isin((2pi)/3)$

So, plugging it all back in:

$(cos (\frac{7 π}{4}) + i sin (\frac{7 π}{4})) - (cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3}))$ $(cos((7pi)/4) + isin((7pi)/4)) - (cos((2pi)/3) + isin((2pi)/3))$

$= (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i) - (- \frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2} i)$ $= (1/sqrt(2) - 1/sqrt(2)i) - (-1/sqrt(2) + sqrt(3)/2i)$

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

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