Question

How do you evaluate `e^( ( 3 pi)/8 i) - e^( ( 19 pi)/6 i)` using trigonometric functions?

Answer 1

`1.249 + 1.424i`

Explanation:

According to Euler's formula,

`e^(ix) = cosx + isinx`.

Using values for `x` from the question,

`x = (3pi)/8`
`e^((3pi)/8i) = cos((3pi)/8) + isin((3pi)/8)`
`= cos67.5 + isin67.5`
`= 0.383 + 0.924i`

`x = (19pi)/6`
`e^((19pi)/6i) = cos((19pi)/6) + isin((19pi)/6)`
`= cos570 + isin570`
`= -0.866 - 0.500i`

Putting these two values together,

`0.383 + 0.924i + 0.866 + 0.500i = 1.249 + 1.424i`

You can check the trigonometric values, there's a margin for error with rounding (or if I've accidentally done radians rather than degrees or vice versa on a calculator) but the equation is correct. Simply use Euler's formula, enter your values for `x` and solve with a calculator.