Euler's formula #e^(i theta) = cos(theta) + i sin(theta)#
Our question #2e^((7pi)/12 i)# can be simplified to using the Euler's formula as
#2(cos((7pi)/12)+ i sin((7pi)/12))#
Now we need to evaluate this.
#cos((7pi)/12) = cos(pi/4 + pi/3)#
#cos((7pi)/12) = cos(pi/4)cos(pi/3) - sin(pi/4)sin(pi/3)#
#cos((7pi)/12) = sqrt(2)/2 * 1/2 - sqrt(2)/2 * sqrt(3)/2#
#cos((7pi)/12) = sqrt(2)/4 - sqrt(6)/4#
#cos((7pi)/12) = (sqrt(2)-sqrt(6))/4#
#sin((7pi)/12) = sin(pi/4 + pi/3)#
#sin((7pi)/12) = sin(pi/4)cos(pi/3) + cos(pi/4)sin(pi/3)#
#sin((7pi)/12) = sqrt(2)/2*1/2 +sqrt(2)/2*sqrt(3)/2#
#sin((7pi)/12) = sqrt(2)/4 + sqrt(6)/4#
#sin((7pi)/12) = (sqrt(2)+sqrt(6))/4#
The complex number would be
#2( (sqrt(2)-sqrt(6))/4 + i (sqrt(2)+sqrt(6))/4)#
#1/2 ( (sqrt(2)-sqrt(6)) + i (sqrt(2)+sqrt(6)))#
That should do for an answer, further simplification is possible depending on how the answer needs to be represented.
