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Chapter 7: Problem 96

If you are given a complex number in rectangular form, how do you write it in polar form?

Short Answer

Expert verified
The complex number \(3 + 4i\) in polar form is \(5(cos(53.13掳) + i sin(53.13掳))\.

Step by step solution

01

Identify Rectangular Form

In rectangular form, a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, \(i\) is the imaginary unit, \(a\) is the real part, and \(b\) is the imaginary part. For example, if a complex number in rectangular form is given as \(3+4i\), \(a = 3\) and \(b = 4\).
02

Calculate Magnitude (absolute value/radius) of Complex Number

In polar form, a complex number is written as \(r(cos(\theta) + \isin(\theta))\), where \(r\) is the magnitude, and \(\theta\) is the angle. The magnitude of a complex number is given by the formula \(\sqrt{a^2 + b^2}\). With \(a=3\) and \(b=4\), the magnitude \(r\) is then calculated as \(r=\sqrt{3^2 + 4^2} = 5\).
03

Calculate Angle (argument/phase) of Complex Number

The angle \(\theta\) is calculated using the inverse tangent or, arctan, function: \(\theta = arctan(b/a)\). So in our example, \(\theta = arctan(4/3)\) or \(\theta \approx 53.13掳\), depending on whether the result is desired in degrees or radians.
04

Write Polar Form

After finding the magnitude and angle, plug in these values into the polar form: \(r(cos(\theta) + \isin(\theta))\). Using our calculated values, the polar form is \(5(cos(53.13掳) + i sin(53.13掳))\).

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