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Chapter 6: Problem 25

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$2-i \sqrt{3}$$

Short Answer

Expert verified
The complex number \(2 - i\sqrt{3}\) in polar form is \(2(\cos(330^掳) + i \sin(330^掳))\) or \(2(\cos(-30^掳) + i \sin(-30^掳))\), depending on whether you are using degrees or radians. Note that 330^掳 is equivalent to -30^掳.

Step by step solution

01

Plotting the complex number

Plot the point (2, -\(\sqrt{3}\)) on the complex plane. The real part, 2, is represented on the x-axis while the imaginary part, -\(\sqrt{3}\), is represented on the y-axis. The point is in the lower right quadrant.
02

Converting to polar form

The polar form of a complex number is given by \(r(cos\(\theta\) + isin\(\theta\))\), where \(r\) is the magnitude and \(\theta\) is the argument or angle from the positive x-axis (expressed in degrees or radians). To find \(r\), use the formula \(r = \sqrt{a^2 + b^2}\) where \(a\) and \(b\) are the real and imaginary parts respectively. So, \(r = \sqrt{2^2 + (\(-\sqrt{3}\))^2} = 2\). To find \(\theta\), use the formula \(\theta = \arctan(\(b/a\))\). So, \(\theta = \arctan(\(-\sqrt{3}/2\))\). Since the point is in the lower right quadrant, we add 360 (or 2\(\pi\), depending on whether you are using degrees or radian measure) to the result of \(\arctan\) to get the correct angle.
03

Writing the polar form

After calculating \(r\) and \(\theta\), write the complex number in polar form as \(r(cos\(\theta\) + i sin\(\theta\)).\)

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Most popular questions from this chapter

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$5 \mathbf{u} \cdot(3 \mathbf{v}-4 \mathbf{w})$$

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Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=3 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}+5 \mathbf{j}$$

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