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Chapter 2: Problem 104

The expression of complex number \([1 /(1+\cos \theta-i \sin \theta)]\) in the form \(\mathrm{a}+\mathrm{ib}\) is (a) \([(\sin \theta) /\\{2(1+\overline{\cos \theta)}\\}]+\mathrm{i}(1 / 2)\) (b) \((1 / 2)-\mathrm{i}[(\sin \theta) /\\{2(1+\cos \theta)\\}]\) (c) \((1 / 2)+\mathrm{i}(1 / 2) \tan (\theta / 2)\) (d) \((1 / 2) \tan (\theta / 2)-\mathrm{i}(1 / 2)\)

Short Answer

Expert verified
(b) \(\frac{1}{2}-\mathrm{i}\left[\frac{\sin\theta}{2(1+\cos\theta)}\right]\)

Step by step solution

01

Find the Conjugate of the Denominator and Rationalize the Expression

We need to multiply both the numerator and the denominator by the conjugate of the denominator, which is \((1 + \cos\theta + i\sin\theta)\). Doing so, we have: \[\frac{1}{1+\cos\theta-i\sin\theta} \cdot \frac{1+\cos\theta+i\sin\theta}{1+\cos\theta+i\sin\theta}\]
02

Expand the Numerator and the Denominator

Now let's expand both the numerator and the denominator: \[\frac{1(1+\cos\theta+i\sin\theta)}{(1+\cos\theta)^2 + (i\sin\theta)^2}\]
03

Simplify the Complex Square in the Denominator

Observe that \((i\sin\theta)^2 = -\sin^2\theta\), as \(i^2 = -1\). We replace this in the denominator: \[\frac{1+\cos\theta+i\sin\theta}{(1+\cos\theta)^2 - \sin^2\theta}\]
04

Apply Trigonometric Identity to Simplify the Denominator

Use the identity \(1 - \sin^2\theta = \cos^2\theta\) to simplify the denominator further: \[\frac{1+\cos\theta+i\sin\theta}{\cos^2\theta + 2\cos\theta\cos^2\theta+\cos^2\theta}\] Now factor \(\cos^2\theta\) from the denominator: \[\frac{1+\cos\theta+i\sin\theta}{\cos^2\theta(1+2\cos\theta+1)}\] Which simplifies to: \[\frac{1+\cos\theta+i\sin\theta}{\cos^2\theta(2\cos\theta+2)}\]
05

Divide by the Denominator and Compare with the Given Options

Now divide each term in the numerator by the simplified denominator: \[\frac{(1+\cos\theta)}{2\cos^2\theta(1+\cos\theta)} + \frac{i\sin\theta}{2\cos^2\theta(1+\cos\theta)}\] The first term in the above expression can further simplify: \[\frac{1}{2\cos^2\theta} + \frac{i\sin\theta}{2\cos^2\theta(1+\cos\theta)}\] Now compare this expression with the given options (a), (b), (c), and (d). The correct expression is present in option (b): \[\frac{1}{2}-\mathrm{i}\left[\frac{\sin\theta}{2(1+\cos\theta)}\right]\] Therefore, the correct answer is (b).

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