Question

How do you evaluate `e^( ( 7 pi)/4 i) - e^( ( 5 pi)/6 i)` using trigonometric functions?

Answer 1

=- 2sin15(cos15+isin15)

Explanation:

=`cos ((7pi)/4) +i sin ((7pi)/4) - cos( (5pi)/6) - i sin ((7pi)/6)`

= `cos ((7pi)/4) -cos ((5pi)/6) +i [ sin((7pi)/4) - sin((5pi)/6)]`

= `-2 sin((31pi)/12) sin ((11pi)/12) +2i sin((11pi)/12) cos ((31pi)/12)`

=`2sin ((11pi)/12)[ -sin((31pi)/12) +i cos((31pi)/12)]`

=`2sin ((11pi)/12)[ -sin((7pi)/12) +i cos((7pi)/12)]`

=2 sin (pi/12)[ -sin105+ i cos 105]

= 2 sin 15(-cos 15 - i sin 15)

=- 2sin15(cos15+isin15)