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Chapter 6: Problem 96

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

Short Answer

Expert verified
For a complex number \(a + bi\) in rectangular form, it can be written in polar form by calculating its modulus \( r = \sqrt{a^2 + b^2} \) and its argument \(\theta\) using trigonometric principles, and then writing it in the form \( r(\cos \theta + i \sin \theta) \).

Step by step solution

01

Calculate the modulus

Calculate the modulus (magnitude) of the complex number. This is calculated using Pythagoras' theorem on the real and imaginary parts of the complex number. That is, \( r = \sqrt{a^2 + b^2} \). This gives the distance of the complex number from the origin on the complex plane.
02

Calculate the argument

To calculate the argument of \(a + bi\), one can use the formulas \(\theta = \arctan(\frac{b}{a})\) when \(a > 0\), and \(\theta = \arctan(\frac{b}{a}) + \pi\) when \(a < 0\). This gives the angle the complex number makes with the positive real axis, measured counterclockwise in radians. For \(a=0\), if \(b>0\) then \(\theta =\frac{\pi}{2}\), if \(b<0\) then \(\theta =-\frac{\pi}{2}\).
03

Write the complex number in polar form

At this point, you know the magnitude \(r\) and the argument \(\theta\) for the complex number. It can now be written in polar form as \(r(\cos \theta + i \sin \theta)\). Note that for angles that are multiples of \(\frac{\pi}{2}\) we will use instead the values \(\cos 0 = 1\), \(\cos \pi = -1\), \(\sin \frac{\pi}{2} = 1\) and \(\sin -\frac{\pi}{2} = -1\) for simplification.

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