Question

How can you use trigonometric functions to simplify `2 e^( ( 4 pi)/3 i )` into a non-exponential complex number?

Answer 1

`-1-isqrt3`

Explanation:

Using `color(blue)" Euler's relationship "`

`re^(itheta) = r( costheta + isintheta )`

`rArr 2e^((4pi)/3 i) = 2 [(cos((4pi)/3) + isin((4pi)/3)]`
`"----------------------------------------------------------"`

now : `cos((4pi)/3) = -cos(pi/3)`

and `sin((4pi)/3) = -sin(pi/3)`

Using `color(red)" exact value triangle "`

From the triangle we can obtain
`-cos(pi/3) = -1/2" and -sin(pi/3) = -sqrt3/2`
`rArr 2e^((4pi)/3 i )= 2( -1/2 -i sqrt3/2) = -1 - isqrt3`