# What is the polar form of #( -4,32 )#?

The rectangular point

Polar points are in the form

To find

Thus,

#r=sqrt((-4)^2+(32)^2)=sqrt(4^2+32^2)=sqrt(4^2+4^2(8^2))=sqrt(4^2(1+8^2))=4sqrt65#

Even though the point

To find

Looking at the image, we have

#tantheta="opposite"/"adjacent"=y/x#

Solving for

#theta=tan^-1(y/x)#

Using our known values:

#theta=tan^-1(32/(-4))=tan^-1(-8)=-1.44644133#

Note, however, that this is a negative value and that

To find the value of this angle in Quadrant *minus* the magnitude of the angle we determined.

That is,

#theta=pi-1.44644133=1.69515132#

So, our point is:

#(r,theta)=(4sqrt65,1.69515132)#

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The polar form of the complex number (-4, 32) is 4∠32°.

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