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

1 Answer
A08
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

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