Question

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

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

Step-by-step solution

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: \[\frac{1}{1+\cos\theta-i\sin\theta} \cdot \frac{1+\cos\theta+i\sin\theta}{1+\cos\theta+i\sin\theta}\]

Expand the Numerator and the Denominator

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

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: \[\frac{1+\cos\theta+i\sin\theta}{(1+\cos\theta)^2 - \sin^2\theta}\]

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: \[\frac{1+\cos\theta+i\sin\theta}{\cos^2\theta(2\cos\theta+2)}\]

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).

Key concepts

Trigonometric Identities

Trigonometric identities are useful mathematical tools that relate the angles and sides of right-angled triangles.
These identities help simplify expressions involving trigonometric functions like sine, cosine, and tangent. In this context, we use the identity

• \( 1 - \sin^2\theta = \cos^2\theta \)

This identity is applied to simplify complicated expressions in the denominator.
It's important to grasp how these identities work because they frequently appear in complex number manipulations.
By transforming terms in the expression, we can manage calculations more effectively, revealing simpler forms.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any irrational or complex numbers from the denominator.
This is achieved by multiplying the numerator and the denominator by the conjugate of the denominator.
In this exercise, we multiply by

• \( 1 + \cos\theta + i\sin\theta \)

This technique clears away the imaginary parts or roots, leaving behind a simpler rational number.
Rationalizing helps in clear calculations and is a standard procedure in complex number algebra.
It evolves expressions into a format that's easier to interpret and compare.

Conjugate in Complex Numbers

The conjugate of a complex number is obtained by changing the sign of the imaginary part.
If a complex number is written as \( a + ib \), its conjugate is \( a - ib \).
Using the conjugate plays a crucial role in rationalizing denominators and simplifying expressions.
In this exercise, the conjugate

• \( 1 + \cos\theta + i\sin\theta \)

helps eliminate the imaginary unit

• \( i \)

in the denominator.
Multiplying by the conjugate transforms products of complex numbers into real numbers, facilitating easier division.

Expressions in a+ib form

Expressing complex numbers in

• \( a + ib \)

form is crucial for easy comparison and computation.
Here, \( a \) and \( b \) are real numbers where \( i \) represents the imaginary unit satisfying \( i^2 = -1 \).
The exercise guides us to transform the given expression into this standard form, allowing us to easily see how the real and imaginary parts correspond to options (a), (b), (c), and (d).
Recognizing the outputs as

• \( 1/2 - i[\sin\theta / 2(1+\cos\theta)] \)

we find the expression aligns perfectly with option (b), which aids in understanding and verifying the solution.