Write the polar form of the complex number z given by z=6-8i.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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))

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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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