How do you evaluate e^( ( pi)/2 i) - e^( ( 23 pi)/3 i) using trigonometric functions?

1 Answer
Dwight
Dec 17, 2016

With the help of Euler's formula we can take this right down to
0.5 + i (1 + (sqrt3)/2)

Explanation:

Start with Euler's formula: e^(ix) = cos x + i Sin x

Your expression becomes:

cos pi/2 + i sin pi/2 - cos (23pi)/3 - i sin (23pi)/3

To simplify, note cos pi/2 = 0, sin pi/2 = 1

and since each complete revolution of a unit circle is 2pi, which we will write as (6pi)/3, cos (23pi)/3 = cos (5pi)/3
and likewise for sin (23pi)/3 = sin (5pi)/3

So, we now have the original expression simplified down to

i - cos (5pi)/3 - i sin (5pi)/3

where cos (5pi)/3 = 0.5 and sin (5pi)/3 = - (sqrt3/2)

So, finally, the result is

0.5 + i (1 + (sqrt3)/2)

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