Question

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

Answer 1

`2/sqrt2`

Explanation:

`e^((pi/4)i`= `cos (pi/4) +i sin (pi/4)= (1+i)/sqrt2`

Like wise,

`e^((3pi)/4i)` = `cos ((3pi)/4) +i sin((3pi)/4)`

=`- cos (pi/4) +i sin (pi/4)`

= `(-1+i)/sqrt 2`

Hence `e^((pi/4)i` `-e^((3pi)/4i)`= `(1+i)/sqrt2 - (-1+i)/sqrt 2`= `2/sqrt2`