How can you use trigonometric functions to simplify $3 e^{\frac{5 π}{3} i}$ $3 e^( ( 5 pi)/3 i )$ into a non-exponential complex number?

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Answer 1 · 2016-10-09T19:47:18

$3 e^{(\frac{5 π}{3}) i} = 3 cos (\frac{5 π}{3}) + 3 sin (\frac{5 π}{3}) i$ $3e^(((5pi)/3)i) = 3cos((5pi)/3) + 3sin((5pi)/3)i$

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

Multiply by any magnitude, A:

$A e^{i x} = A cos (x) + A sin (x) i$ $Ae^(ix) = Acos(x) + Asin(x)i$

Answer 2 · 2016-10-09T19:52:00

$\frac{- \sqrt{3} + i}{2}$ $(-sqrt3+i)/2$

Using Euler's formula $e^{i x} = sin x + i cos x$ $e^(ix)=sinx+icosx$ ...

$3 e^{\frac{5 π}{3} i} = sin (\frac{5 π}{3}) + i cos (\frac{5 π}{3})$ $3e^[(5pi)/3i]=sin((5pi)/3)+icos((5pi)/3)$

$a a a a = - \frac{\sqrt{3}}{2} + i * \frac{1}{2} a a a$ $color(white)(aaaa)=-sqrt3/2+i**1/2color(white)(aaa)$ from the unit circle

$a a a a = \frac{- \sqrt{3} + i}{2}$ $color(white)(aaaa)=(-sqrt3+i)/2$

How can you use trigonometric functions to simplify 3e5π3i 3 e^( ( 5 pi)/3 i ) into a non-exponential complex number?