Question

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

Answer 1

`sin ( pi)/4 + i cos ( pi)/4`
- `sin ( 4 pi)/3 + i cos ( 4 pi)/3`

Explanation:

In complex terms, `R ( sintheta + i costheta )`= `Re ^ (i theta)`
where `R` is the modulus and `theta` is the argument.

So
`e^( ( pi)/4 i) - e^( ( 4 pi)/3 i)` = `1 ( sin ( pi)/4 + i cos ( pi)/4 )`
- `1 ( sin ( 4 pi)/3 + i cos ( 4 pi)/3 )`

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