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Chapter 6: Problem 78

Convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$(1+i)(1-i \sqrt{3})(-\sqrt{3}+i)$$

Short Answer

Expert verified
The result in polar form is \(2\sqrt{2}(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6})\). In rectangular form, the result is \(-\sqrt{6} + i\sqrt{2}\).

Step by step solution

01

Convert to Polar Form

The first step to solving this exercise is to convert each of the complex numbers to their polar forms. To do this, recall that a complex number \(a + bi\) can be written in polar form as \(r(\cos\theta + i\sin\theta)\), where \(r = \sqrt{a^2+b^2}\) and \(\theta = \tan^{-1}(\frac{b}{a})\). For \(1 + i\), \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\). For \(1 - i\sqrt{3}\), \(r = 2\) and \(\theta = -\frac{\pi}{3}\). For \(-\sqrt{3} + i\), \(r = 2\) and \(\theta = \frac{2\pi}{3}\). Therefore, the polar forms are \(\sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})\), \(2(\cos-\frac{\pi}{3} + i\sin-\frac{\pi}{3})\), and \(2(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3})\).
02

Perform Operations

Next, perform the multiplication in the polar forms. This is simpler than it would be in rectangular form. In polar form, multiply the magnitudes and add the angles. This gives \(2\sqrt{2}\left(\cos\left(\frac{\pi}{4} - \frac{\pi}{3} + \frac{2\pi}{3}\right) + i\sin\left(\frac{\pi}{4} - \frac{\pi}{3} + \frac{2\pi}{3}\right)\right)\).
03

Calculate Polar Form

Calculate the resultant polar form by simplifying the expression obtained in the previous step. This gives \(2\sqrt{2}(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6})\). This is the result in the polar form.
04

Convert to Rectangular Form

Finally, convert back to rectangular form to express the final answer in rectangular form. To do this, multiply the magnitude with \(\cos\theta\) for the real part and \(\sin\theta\) for the imaginary part. This gives \(2\sqrt{2}(\frac{-\sqrt{3}}{2}) + 2\sqrt{2}(\frac{1}{2})i = -\sqrt{6} + i\sqrt{2}\).

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