How can you use trigonometric functions to simplify $15 e^{\frac{π}{3} i}$ $15 e^( ( pi)/3 i )$ into a non-exponential complex number?

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Answer 1 · 2016-05-16T00:30:28

$15 e^{\frac{π}{3} i} = \frac{15}{2} + \frac{15 \sqrt{3}}{2} i$ $15e^(pi/3i)=15/2+(15sqrt(3))/2i$

We will use Euler's formula:

$e^{i θ} = cos (θ) + i sin (θ)$ $e^(i theta) = cos(theta)+isin(theta)$

With that, we have

$15 e^{\frac{π}{3} i} = 15 (cos (\frac{π}{3}) + i sin (\frac{π}{3}))$ $15e^(pi/3i) = 15(cos(pi/3)+isin(pi/3))$

$= 15 (\frac{1}{2} + \frac{\sqrt{3}}{2} i)$ $=15(1/2+sqrt(3)/2i)$

$= \frac{15}{2} + \frac{15 \sqrt{3}}{2} i$ $=15/2+(15sqrt(3))/2i$

How can you use trigonometric functions to simplify 15eπ3i 15 e^( ( pi)/3 i ) into a non-exponential complex number?