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

1 Answer
Feb 21, 2016

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