How can you use trigonometric functions to simplify # 4 e^( ( 5 pi)/4 i ) # into a non-exponential complex number?

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Answer 1 · 2016-06-14T20:37:29

The value of this complex number is #-2sqrt(2)-2sqrt(2)i#

To express a complex number without exponents you use the following formula:

#|z|*e^(varphii)=|z|(cosvarphi+isinvarphi)#

In the example above you have:

#|z|=4#

#varphi=(5pi)/4#

So you have:

#4e^((5pi)/4*i)=4*(cos((5pi)/4)+i*sin((5pi)/4))=#

#4*(cos(pi+pi/4)+i*sin(pi+pi/4))=#
#4*(-cos(pi/4)-isin(pi/4))=#
#4*(-sqrt(2)/2-i*sqrt(2)/2)=-2sqrt(2)-2sqrt(2)*i#