How do you evaluate eπ4ie11π8i using trigonometric functions?

1 Answer
Aug 13, 2018

The answer is =22222+i(222+22)

Explanation:

Apply Euler's Identity

eiθ=cosθ+isinθ

eiπ4=cos(π4)+isin(π4)

=22+i22

ei118π=cos(118π)+isin(118π)

cos(2θ)=2cos2θ1

cosθ=1+cos2θ2

cos(118π)= (1+cos(114π)2)

=1222

=222

cos(2θ)=12sin2θ

sinθ=1cos(2θ)2

sin(118π)=1cos(114π)2

=1+222

=2+22

Finally,

eiπ4ei118π

=22+i22222i2+22

=22222+i(222+22)