How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{3 π}{8} i}$ $e^( ( pi)/4 i) - e^( ( 3 pi)/8 i)$ using trigonometric functions?

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{3 π}{8} i}$ $e^( ( pi)/4 i) - e^( ( 3 pi)/8 i)$ using trigonometric functions?

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Answer 1 · 2016-03-23T02:47:11

$e^{\frac{π}{4} i} - e^{\frac{3 π}{8} i} \approx 0.324 - i 0.217$ $e^(pi/4 i) -e^((3pi)/8 i) ~~0.324-i0.217$

Explanation:

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

$θ_{1} = \frac{π}{4}, θ_{2} = \frac{3 π}{8}$ $theta_1=pi/4, theta_2 = (3pi)/8$

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

$e^{\frac{3 π}{8} i} = cos (\frac{3 π}{8}) + i sin (\frac{3 π}{8})$ $e^((3pi)/8 i) = cos( (3pi)/8) + i sin ((3pi)/8)$

$e^{\frac{π}{4} i} - e^{\frac{3 π}{8} i} = [cos (\frac{π}{4}) + i sin (\frac{π}{4})] - [cos (\frac{3 π}{8}) + i sin (\frac{3 π}{8})]$ $e^(pi/4 i) -e^((3pi)/8 i) = [cos (pi/4) +isin(pi/4)] - [cos( (3pi)/8) + i sin ((3pi)/8)]$

$e^{\frac{π}{4} i} - e^{\frac{3 π}{8} i} = [cos (\frac{π}{4}) - cos (\frac{3 π}{8})] - i [sin (\frac{π}{4}) + sin (\frac{3 π}{8})]$ $e^(pi/4 i) -e^((3pi)/8 i) = [cos (pi/4) -cos( (3pi)/8)] - i[ sin(pi/4)+sin ((3pi)/8)]$

$\approx 0.324 - i 0.217$ $~~0.324-i0.217$

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{3 π}{8} i}$ $e^( ( pi)/4 i) - e^( ( 3 pi)/8 i)$ using trigonometric functions?

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