Question

How do you evaluate `e^( ( 13 pi)/8 i) - e^( (5 pi)/12 i)` using trigonometric functions?

Answer 1

`e^((13pi)/8i)-e^((5pi)/12i)~~0.12-1.89i`

Explanation:

We can represent `ae^(ix)` in trig form as `ae^(ix)=a(cosx+isinx)`

Using this for `e^((13pi)/8i)-e^((5pi)/12i)` gives us:
`(cos((13pi)/8)+isin((13pi)/8))-(cos((5pi)/12)+isin((5pi)/12))`

`=cos((13pi)/8)+isin((13pi)/8)-cos((5pi)/12)-isin((5pi)/12))`

`=cos((13pi)/8)-cos((5pi)/12)-isin((5pi)/12)+isin((13pi)/8)`

`=(cos((13pi)/8)-cos((5pi)/12))-i(sin((5pi)/12)-sin((13pi)/8))`

`~~0.1238643873-i(1.889805359)`

`=0.1238643873-1.889805359i`

`~~0.12-1.89i`