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Chapter 7: Problem 95

What is the polar form of a complex number?

Short Answer

Expert verified
The polar form of a complex number \(a + bi\) is calculated by finding the magnitude \(r = \sqrt{a^2 + b^2}\) and the angle \(\theta = tan^{-1}\left(\frac{b}{a}\right)\), resulting in the form \(r(cos \(\theta\) + isin\(\theta\)) or r \(\text{cis}\) \(\theta\).

Step by step solution

01

Calculate the magnitude of the complex number

To calculate the magnitude of any complex number, \(a + bi\), use Pythagoras' theorem. The magnitude \(r\) is obtained by \(r = \sqrt{a^2 + b^2}\). This represents the distance of the point \((a, b)\) from the origin in a 2D space.
02

Calculate the angle

The angle \(\theta\) can be calculated using \(\theta = tan^{-1}\left(\frac{b}{a}\right)\), as \(b/a\) represents the tangent of the angle formed with the real axis. Use the appropriate quadrant depending on the signs of \(a\) and \(b\).
03

Writing in the polar form

After determining \(r\) and \(\theta\), substitute these values into the polar form \(r(cos \(\theta\) + isin\(\theta\)) or r \(\text{cis}\) \(\theta\). This converts the complex number from rectangular form to polar form.

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Most popular questions from this chapter

Use a graphing utility to graph the polar equation. $$r=4 \cos 6 \theta$$

Find the smallest interval for \(\theta\) starting with \(\theta \min =0\) so that your graphing utility graphs the given polar equation exactly once without retracing any portion of it. $$r=4 \sin \theta$$

Exercises \(119-121\) will help you prepare for the material covered in the next section. Find the obtuse angle \(\theta,\) rounded to the nearest tenth of a degree, satisfying. If \(w=-2 i+6 j,\) find the following vector: $$ \frac{2(-2)+4(-6)}{|\mathbf{w}|^{2}} \mathbf{w} $$

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

Graph the spiral \(r=\theta .\) Use a \([-48,48,6]\) by \([-30,30,6]\) viewing rectangle. Let \(\theta\) min \(=0\) and \(\theta \max =2 \pi,\) then \(\theta \min =0\) and \(\theta \max =4 \pi,\) and finally \(\theta \min =0\) and \(\theta \max =8 \pi\)
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