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Chapter 6: Problem 11

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$2+2 i$$

Short Answer

Expert verified
The polar form of the complex number '2+2i' is \(2\sqrt{2} (\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))\). It is plotted in the complex plane at point (2,2).

Step by step solution

01

Plotting

Plot the complex number on a complex plane. The x-coordinate represents the real part and the y-coordinate represents the imaginary part. So, '2+2i' should be plotted at point (2,2).
02

Compute the magnitude

Compute the magnitude (r) using the Pythagorean Theorem. The magnitude r = \(\sqrt{\text{{real part}}^2 + \text{{imaginary part}}^2}\) = \(\sqrt{2^2 + 2^2}\) = \(\sqrt{8}\) = \(2\sqrt{2}\). Note that the magnitude can also be thought of as the length of the line from the origin to the point (2,2).
03

Compute the argument

To find the argument \(\theta\), use the formula \(\theta = \text{{atan2}}(\text{{imaginary part}}, \text{{real part}})\). Therefore, \(\theta = \text{{atan2}}(2, 2)\) which equals \(\frac{\pi}{4}\) in radians or 45 degrees.
04

Write the polar form

Combine the magnitude and argument to express the complex number in polar form. Thus, the polar form is \(r(\cos \theta + i\sin \theta) = 2\sqrt{2} (\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))\).

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