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Chapter 2: Problem 143

\(\left[\left\\{(\cos 2 \theta-\mathrm{i} \sin 2 \theta)^{7}(\cos 3 \theta+\mathrm{i} \sin 3 \theta)^{-5}\right\\}\right.\) \(/\left\\{(\cos 4 \theta+i \sin 4 \theta)^{12}(\cos 5 \theta+i \sin 5 \theta)^{-6}\right]=\) (a) \(\cos 33 \theta+i \sin 33 \theta\) (b) \(\cos 33 \theta-\mathrm{i} \sin 33 \theta\) (c) \(\cos 47 \theta+i \sin 47 \theta\) (d) \(\cos 47 \theta+\mathrm{i} \sin 47 \theta\)

Short Answer

Expert verified
(d) \(\cos 29\theta + \mathrm{i} \sin 29\theta\)

Step by step solution

01

Identify the components of the expression

The given expression can be broken down into several components, which are complex numbers in the polar form raised to some power. Let's identify those components: 1. \((\cos 2\theta - \mathrm{i} \sin 2\theta)^7\) 2. \((\cos 3\theta + \mathrm{i} \sin 3\theta)^{-5}\) 3. \((\cos 4\theta + \mathrm{i} \sin 4\theta)^{12}\) 4. \((\cos 5\theta + \mathrm{i} \sin 5\theta)^{-6}\)
02

Apply De Moivre's Theorem to each component

Using De Moivre's theorem, we can now rewrite each of the components above: 1. \[\left(\cos 2\theta - \mathrm{i} \sin 2\theta\right)^7 = \cos (14\theta) - \mathrm{i} \sin (14\theta)\] 2. \[\left(\cos 3\theta + \mathrm{i} \sin 3\theta\right)^{-5} = \cos (-15\theta) + \mathrm{i} \sin (-15\theta)\] 3. \[\left(\cos 4\theta + \mathrm{i} \sin 4\theta\right)^{12} = \cos (48\theta) + \mathrm{i} \sin (48\theta)\] 4. \[\left(\cos 5\theta + \mathrm{i} \sin 5\theta\right)^{-6} = \cos (-30\theta) - \mathrm{i} \sin (-30\theta)\]
03

Multiply the components

Now, multiply components 1 and 2 together, and components 3 and 4 together: \[\frac{(\cos 14\theta - \mathrm{i} \sin 14\theta)(\cos (-15\theta) + \mathrm{i} \sin (-15\theta)}{(\cos 48\theta + \mathrm{i} \sin 48\theta)(\cos (-30\theta) - \mathrm{i} \sin (-30\theta))}\]
04

Use angle subtraction formulas to simplify

To simplify the expression, we will use the angle subtraction formulas: \[\cos (a - b) = \cos a \cos b + \sin a \sin b\] \[\sin (a - b) = \sin a \cos b - \cos a \sin b\] Now, substitute \(a = 14\theta\) and \(b = -15\theta\): \[\cos (14\theta + 15\theta) = \cos 14\theta \cos 15\theta - \sin 14\theta \sin 15\theta\] \[\sin (14\theta + 15\theta) = \sin 14\theta \cos 15\theta + \cos 14\theta \sin 15\theta\] Finally, substitute \(a = 48\theta\) and \(b = -30\theta\): \[\cos (48\theta + 30\theta) = \cos 48\theta \cos 30\theta - \sin 48\theta \sin 30\theta\] \[\sin (48\theta + 30\theta) = \sin 48\theta \cos 30\theta + \cos 48\theta \sin 30\theta\]
05

Combine the results

Now, we can combine our results from Steps 2-4 to get the final simplified expression: \(\cos (29\theta) + \mathrm{i} \sin (29\theta)\)
06

Identify the correct answer

Comparing this expression with the given options, we see that the correct answer is: (d) \(\cos 29\theta + \mathrm{i} \sin 29\theta\)

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