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Chapter 6: Problem 16

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$-3 i$$

Short Answer

Expert verified
The polar form of the complex number -3i is \( 3 (\cos 270^{\circ} + i \sin 270^{\circ}) \) or \( 3 (\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) \) in radians.

Step by step solution

01

Identify the complex number

The given complex number is \(-3i\). Here, the real part is 0, and the imaginary part is -3.
02

Plot the complex number

To plot this complex number on the complex plane, mark a point on the negative y-axis at -3 (this represents the imaginary part). The real part is 0, so the point lies on the origin in the x-direction.
03

Convert to polar form

The polar form of a complex number is given by \( r (\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (absolute value) and \( \theta \) is the argument of the complex number. For \( -3i \), \( r \) equals to 3 (the absolute value of -3) and the angle \( \theta \) it makes with the positive x-axis as 270 degrees or \( \frac{3\pi}{2} \) radians, since the point lies in the fourth quadrant which is counterclockwise from the positive x-axis. Therefore, the polar form of -3i is \( 3 (\cos 270^{\circ} + i \sin 270^{\circ}) \) or \( 3 (\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) \) in radians.

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Most popular questions from this chapter

Verify the identity: $$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x$$ (Section 5.1, Example 5)

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Will help you prepare for the material covered in the next section. Refer to Section 2.1 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i\). $$(1+i)(2+2 i)$$
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