How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{11 π}{8} i}$ $e^( ( pi)/4 i) - e^( ( 11 pi)/8 i)$ using trigonometric functions?

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{11 π}{8} i}$ $e^( ( pi)/4 i) - e^( ( 11 pi)/8 i)$ using trigonometric functions?

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Answer 1 · 2018-08-13T13:34:13

The answer is $= \frac{\sqrt{2}}{2} - \frac{\sqrt{2 - \sqrt{2}}}{2} + i (\frac{\sqrt{2}}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2})$ $=sqrt2/2-sqrt(2-sqrt2)/2+i(sqrt2/2-sqrt(2+sqrt2)/2)$

Explanation:

Apply Euler's Identity

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

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

$= \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$ $=sqrt2/2+isqrt2/2$

$e^{i \frac{11}{8} π} = cos (\frac{11}{8} π) + i sin (\frac{11}{8} π)$ $e^(i11/8pi)=cos(11/8pi)+isin(11/8pi)$

$cos (2 θ) = 2 {cos}^{2} θ - 1$ $cos(2theta)=2cos^2theta-1$

$cos θ = \sqrt{\frac{1 + cos 2 θ}{2}}$ $costheta=sqrt((1+cos2theta)/2)$

$cos (\frac{11}{8} π) = \sqrt{(1 + \frac{cos (\frac{11}{4} π)}{2})}$ $cos(11/8pi)=sqrt((1+cos(11/4pi)/2)$

$= \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}$ $=sqrt((1-sqrt2/2)/2)$

$= \frac{\sqrt{2 - \sqrt{2}}}{2}$ $=sqrt(2-sqrt2)/2$

$cos (2 θ) = 1 - 2 {sin}^{2} θ$ $cos(2theta)=1-2sin^2theta$

$sin θ = \sqrt{\frac{1 - cos (2 θ)}{2}}$ $sintheta=sqrt((1-cos(2theta))/2)$

$sin (\frac{11}{8} π) = \sqrt{\frac{1 - cos (\frac{11}{4} π)}{2}}$ $sin(11/8pi)=sqrt((1-cos(11/4pi))/2)$

$= \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ $=sqrt((1+sqrt2/2)/2)$

$= \frac{\sqrt{2 + \sqrt{2}}}{2}$ $=sqrt(2+sqrt2)/2$

Finally,

$e^{i \frac{π}{4}} - e^{i \frac{11}{8} π}$ $e^(ipi/4)-e^(i11/8pi)$

$= \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} - \frac{\sqrt{2 - \sqrt{2}}}{2} - i \frac{\sqrt{2 + \sqrt{2}}}{2}$ $=sqrt2/2+isqrt2/2-sqrt(2-sqrt2)/2-isqrt(2+sqrt2)/2$

$= \frac{\sqrt{2}}{2} - \frac{\sqrt{2 - \sqrt{2}}}{2} + i (\frac{\sqrt{2}}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2})$ $=sqrt2/2-sqrt(2-sqrt2)/2+i(sqrt2/2-sqrt(2+sqrt2)/2)$

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{11 π}{8} i}$ $e^( ( pi)/4 i) - e^( ( 11 pi)/8 i)$ using trigonometric functions?

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