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Chapter 6: Problem 15

Write \(2-2 i\) in polar form.

Short Answer

Expert verified
The polar form of the complex number \(2 - 2i\) is \(\sqrt{8}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4})\).

Step by step solution

01

Identify the real and imaginary parts

For the complex number \(2-2i\), the real part \(a\) is \(2\), and the imaginary part \(b\) is \(-2\).
02

Calculate the magnitude of the complex number

We will use the magnitude formula \(r = \sqrt{a^2 + b^2}\). In our case, \(a = 2\) and \(b = -2\). So we have: \[r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8}.\] The magnitude of the complex number is \(\sqrt{8}\).
03

Calculate the angle of the complex number

We will use the angle formula \(\theta = \arctan\left(\frac{b}{a}\right)\). In our case, \(a = 2\) and \(b = -2\). So we have: \[\theta = \arctan\left(\frac{-2}{2}\right) = \arctan(-1).\] Since the complex number is in the third quadrant, we should add \(\pi\) to the angle: \[\theta = \arctan(-1) + \pi = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}.\] The angle of the complex number is \(\frac{3\pi}{4}\).
04

Write the polar form

Now that we have the magnitude and the angle, we can write the polar form of the complex number. The polar form of a complex number is given by: \[z = r(\cos\theta + i\sin\theta)\] In our case, \(r = \sqrt{8}\) and \(\theta = \frac{3\pi}{4}\). So we have: \[2 - 2i = \sqrt{8}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}).\] The polar form of the complex number \(2 - 2i\) is \(\sqrt{8}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4})\).

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Most popular questions from this chapter

Give an example of an angle \(\theta\) such that both \(\sin \theta\) and \(\sin (2 \theta)\) are rational.

Show that $$ |\sin (2 \theta)| \leq 2|\sin \theta| $$ for every angle \(\theta\).

Sketch the graph of the function \(7 \cos \left(\frac{\pi}{2} x+\frac{6 \pi}{5}\right)+3\) on the interval [-8,8]

What is the relationship between the point with polar coordinates \(r=5, \theta=0.2\) and the point with polar coordinates \(r=5, \theta=0.2+\pi ?\)

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=8, \theta=\frac{\pi}{3} $$
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