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

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Answer 1 · 2017-01-10T21:57:14

The value of this expression is #1/2+(1-sqrt(3)/2)i#

To evaluate this expression you have to write the complex numbers in algebraic form. To do this you use the identity:

#e^(varphii)=cos varphi+ isinvarphi#

In the given example we get:

#e^(pi/2i)-e^((2pi)/3i)=(cos(pi/2)+isin(pi/2))-(cos((2pi)/3)+isin((2pi)/3))=#

#=i-(cos(pi-pi/3)+isin(pi-pi/3))=#

#=i-(-cos(pi/3)+isin(pi/3))=#

#=i+cos(pi/3)-isin(pi/3)=#

#i+1/2-sqrt(3)/2*i=1/2+(1-sqrt(3)/2)i#