Express 1/(1 - cos θ + 2i sin θ) in the form x + iy.

Q: Why did we use half-angle identities instead of just multiplying by the conjugate of the original denominator?

Great question! While you *could* multiply by the conjugate of $1 - \cos \theta + 2i \sin \theta$, the resulting algebra would be incredibly messy and difficult to simplify. Using half-angle identities first allows us to factor out a common term, which makes the rationalization process much cleaner and more elegant.

Q: Is it okay to leave the answer in terms of $\theta/2$?

While mathematically correct, it is standard practice in JEE problems to express the final answer in terms of the original variable, $\theta$. By converting back to full angles, we ensure our answer is in the most simplified and standard form.

Q: How do I know when to use $1 - \cos \theta = 2 \sin^2 (\theta/2)$?

Whenever you see $1 \pm \cos \theta$ in a denominator or a complex expression, it is almost always a hint to use half-angle identities. It is a powerful tool to reduce the complexity of trigonometric expressions.

Question

Express $$\frac{1}{1 - \cos 	heta + 2i \sin 	heta}$$ in the form $x + iy$.

Sigma Percentile · Accepted Answer

### The Beauty of Complex Simplification

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to demystify a complex number problem that, at first glance, might seem like a tangled mess of trigonometry and imaginary units.

We are tasked with expressing the following expression in the standard form $x + iy$:

$$Z = \frac{1}{1 - \cos 	heta + 2i \sin 	heta}$$

This is not just an algebraic exercise; it is a lesson in pattern recognition and the art of simplification.

### The Spark of Insight

When you look at the denominator, $1 - \cos 	heta + 2i \sin 	heta$, you should immediately identify the $1 - \cos 	heta$ and $\sin 	heta$ terms as classic triggers for half-angle identities.

We utilize the following identities:

$$1 - \cos 	heta = 2 \sin^2 \frac{	heta}{2}$$

$$\sin 	heta = 2 \sin \frac{	heta}{2} \cos \frac{	heta}{2}$$

By substituting these, we transform our denominator into:

$$2 \sin^2 \frac{	heta}{2} + 4i \sin \frac{	heta}{2} \cos \frac{	heta}{2}$$

Suddenly, the expression is no longer a random collection of terms; it has a structure we can work with.

### The Power of Factoring

Now, consider our new denominator: $2 \sin^2 \frac{	heta}{2} + 4i \sin \frac{	heta}{2} \cos \frac{	heta}{2}$. Both terms share a common factor of $2 \sin \frac{	heta}{2}$.

When we factor this out, we obtain:

$$2 \sin \frac{	heta}{2} \left( \sin \frac{	heta}{2} + 2i \cos \frac{	heta}{2} ight)$$

This is the "Aha!" moment. We have isolated the complex part, making the next step—rationalization—much more straightforward.

### The Rationalization Dance

To eliminate the imaginary unit from the denominator, we multiply the numerator and the denominator by the complex conjugate of the bracketed term, which is $\sin \frac{	heta}{2} - 2i \cos \frac{	heta}{2}$.

The denominator becomes:

$$2 \sin \frac{	heta}{2} \left( \sin^2 \frac{	heta}{2} + 4 \cos^2 \frac{	heta}{2} ight)$$

Using the identity $\sin^2 A + \cos^2 A = 1$, we can simplify $\sin^2 \frac{	heta}{2} + 4 \cos^2 \frac{	heta}{2}$ to $1 + 3 \cos^2 \frac{	heta}{2}$.

### The Final Transformation

We are almost there. The final step is to convert our half-angle terms back to the full angle $	heta$.

Using the identity $2 \cos^2 \frac{	heta}{2} = 1 + \cos 	heta$, we can show that $1 + 3 \cos^2 \frac{	heta}{2} = \frac{5 + 3 \cos 	heta}{2}$. Substituting this back into our expression, the real and imaginary parts separate beautifully.

The **final result** is:

$$\left( \frac{1}{5 + 3 \cos 	heta} ight) + i \left( \frac{-2 \cot \frac{	heta}{2}}{5 + 3 \cos 	heta} ight)$$

This journey from a messy fraction to a clean, elegant form is the essence of mathematics. Keep practicing, keep visualizing, and you will find that even the most intimidating problems have a simple, logical soul.

Animated Solution for Mathematics - Complex Numbers: Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

Visualized Solution

Identify the Goal

Recall Half-Angle Identities

Substitute Identities

Factor the Denominator

Rationalize the Expression

Multiply by Conjugate

Simplify the Denominator

Separate Real and Imaginary Parts

Convert Back to Full Angle

Simplify the Full Angle Term

Final Form $x + i y$

The Sigma Insight: Algebraic Operations on Complex Numbers

The Beauty of Complex Simplification

The Spark of Insight

The Power of Factoring

The Rationalization Dance

The Final Transformation

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Express 1−cosθ+2isinθ1​ in the form x+iy.