Question

How do you evaluate `e^( ( pi)/2 i) - e^( ( 4 pi)/3 i)` using trigonometric functions?

Answer 1

the answer is `=1/2+i(1+sqrt3/2)`

Explanation:

Let `z= e^(ipi/2)-e^(4ipi/3)`
Use the relation `e^(itheta)=costheta + isintheta`
So `e^(ipi/2)=cos(pi/2)+isin(pi/2)=0+i=i`
and `e^(4ipi/3)=cos((4pi)/3)+isin((4pi)/3)=-1/2-isqrt3/2`
so `z=i+1/2+isqrt3/2=1/2+i(1+sqrt3/2)`