Write the polar form of the complex number z given by z=6-8i.
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The polar form of a complex number z = x + yi is r*(cos A + i* sin A)
where tan A= y/x and r = sqrt (x^2 + y^2)
Here we have z = 6 - 8i
r = sqrt( 6^2 + 8^2) = sqrt (36 + 64) = sqrt 100 = 10
tan A = -8/6 = -4/3
A = arc tan (-4/3)
z = 10*( cos (arc tan(-4/3) + i* sin (arc tan(-4/3))
The required polar form of z = 6 - 8i is z = 10*( cos (arc tan(-4/3) + i* sin (arc tan(-4/3))
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The original given form of the complex number is rectangular form.
We'll put the number into the polar form.
z = a + bi
z = 6-8i
Re(z) = 6 and Im(z) = -8
The polar form:
z = |z|(cos t + i sin t)
|z| = sqrt[Re(z)^2 + Im(z)^2]
|z| = sqrt [(6)^2 + (-8)^2]
|z| = sqrt (36 + 64)
|z| = sqrt 100
|z| = 10
tan t = Im (z)/Re(z)
tan t = -8/6
tan t = -4/3
t = arctan(-4/3)
The polar form of the complex number z is: z = 10{cos [arctan(-4/3)] + i*sin [arctan(-4/3)]}.
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