Question

How can you use trigonometric functions to simplify `22 e^( ( 3 pi)/4 i )` into a non-exponential complex number?

Answer 1

`22[cos ((3pi)/4)+j sin ((3pi)/4)]` or
`-15.5562+15.5562j`
See details below.

Explanation:

Following expressions have the same value

`x + yj`, Cartesian or Rectangular form

`r(cos θ+j sin θ) =r cis θ =r/_θ`, Polar form

where `|r|=sqrt(x^2+y^2)`, and
`theta =tan^(-1)(y/x)`
(The angle of the point on the complex plane obtained by taking inverse tangent of the complex portion over the real portion)

`re^(jθ)`, Exponential from.

The question asks us to change from exponential form using trigonometric functions. Therefore, let's chose to change from exponential form to polar form.

`re^(jθ)=r(cos θ+j sin θ)`
`22e^(j(3pi)/4),` Inspection reveals that `r=22and theta =(3pi)/4`
Polar* form becomes `22[cos ((3pi)/4)+j sin ((3pi)/4)]`
Inserting the value of cosine and sine of the angle `(3pi)/4=135^@`we obtain
`22(-0.7071+0.7071j)`

Expressing in rectangular* form

`-15.5562+15.5562j`

*Both Polar Form and Rectangular Form the are required non-exponential complex numbers