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

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Answer 1 · 2018-08-13T13:34:13

The answer is $=sqrt2/2-sqrt(2-sqrt2)/2+i(sqrt2/2-sqrt(2+sqrt2)/2)$

Apply Euler's Identity

$e^(itheta)=costheta+isintheta$

$e^(ipi/4)=cos(pi/4)+isin(pi/4)$

$=sqrt2/2+isqrt2/2$

$e^(i11/8pi)=cos(11/8pi)+isin(11/8pi)$

$cos(2theta)=2cos^2theta-1$

$costheta=sqrt((1+cos2theta)/2)$

$cos(11/8pi)=sqrt((1+cos(11/4pi)/2)$

$=sqrt((1-sqrt2/2)/2)$

$=sqrt(2-sqrt2)/2$

$cos(2theta)=1-2sin^2theta$

$sintheta=sqrt((1-cos(2theta))/2)$

$sin(11/8pi)=sqrt((1-cos(11/4pi))/2)$

$=sqrt((1+sqrt2/2)/2)$

$=sqrt(2+sqrt2)/2$

Finally,

$e^(ipi/4)-e^(i11/8pi)$

$=sqrt2/2+isqrt2/2-sqrt(2-sqrt2)/2-isqrt(2+sqrt2)/2$

$=sqrt2/2-sqrt(2-sqrt2)/2+i(sqrt2/2-sqrt(2+sqrt2)/2)$

How do you evaluate e^( ( pi)/4 i) - e^( ( 11 pi)/8 i) e^( ( pi)/4 i) - e^( ( 11 pi)/8 i) using trigonometric functions?