How can you use trigonometric functions to simplify $25 e^{\frac{2 π}{3} i}$ $25 e^( ( 2 pi)/3 i )$ into a non-exponential complex number?

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How can you use trigonometric functions to simplify $25 e^{\frac{2 π}{3} i}$ $25 e^( ( 2 pi)/3 i )$ into a non-exponential complex number?

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Answer 1 · 2016-03-23T16:09:06

$25 e^{\frac{2 π}{3} i} = - \frac{25}{2} + 25 i \frac{\sqrt{3}}{2}$ $25e^((2pi)/3 i )=-25/2+25isqrt3 /2$

Explanation:

$r e^{i θ} = r (cos θ + i sin θ)$ $re^(itheta)=r(cos theta+i sin theta)$
$25 e^{\frac{2 π}{3} i} \to r = 25, θ = (\frac{2 π}{3})$ $25e^((2pi)/3 i )->r=25, theta=((2pi)/3)$
$25 e^{\frac{2 π}{3} i} = 25 (cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3}))$ $25e^((2pi)/3 i )=25(cos ((2pi)/3) + i sin ((2pi)/3))$
$25 e^{\frac{2 π}{3} i} = 25 cos (\frac{2 π}{3}) + 25 i sin (\frac{2 π}{3})$ $25e^((2pi)/3 i )=25cos ((2pi)/3) + 25i sin ((2pi)/3)$
$25 e^{\frac{2 π}{3} i} = - \frac{25}{2} + 25 i \frac{\sqrt{3}}{2}$ $25e^((2pi)/3 i )=-25/2+25isqrt3 /2$

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How can you use trigonometric functions to simplify $25 e^{\frac{2 π}{3} i}$ $25 e^( ( 2 pi)/3 i )$ into a non-exponential complex number?

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How can you use trigonometric functions to simplify $25 e^{\frac{2 π}{3} i}$ $25 e^( ( 2 pi)/3 i )$ into a non-exponential complex number?