Problem 27 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\tan \beta=2 \sin \alpha \cdot \sin \gamma \cdot \operatorname{cosec}(\alpha+\gamma)\), then \(\cot \alpha, \cot \beta, \cot \gamma\) are in (a) A.P (b) G.P (c) H.P (d) A.G.P

Step-by-Step Solution

Answer

(a) A.P

Step 1: Understand and express the given condition

The given condition is \(\tan \beta = 2 \sin \alpha \cdot \sin \gamma \cdot \csc(\alpha + \gamma)\). We know that \(\csc(\alpha + \gamma) = \frac{1}{\sin(\alpha + \gamma)}\). Thus we simplify the equation: \(\tan \beta = 2 \sin \alpha \cdot \sin \gamma \cdot \frac{1}{\sin(\alpha + \gamma)}\)

Step 2: Simplify the equation

The right-hand side of the equation becomes \(2 \sin \alpha \cdot \sin \gamma \cdot \frac{1}{\sin(\alpha + \gamma)} = 2 \frac{\sin \alpha \cdot \sin \gamma}{\sin(\alpha + \gamma)}\). This can be further simplified using the sine addition formula: \(\sin(\alpha + \gamma) = \sin \alpha \cos \gamma + \cos \alpha \sin \gamma\). Thus our equation simplifies to: \(\tan \beta = \frac{2 \sin \alpha \cdot \sin \gamma}{\sin \alpha \cos \gamma + \cos \alpha \sin \gamma}\). By dividing numerator and denominator by \(\sin\alpha sin\gamma\), we get: \(\tan \beta = \frac{2}{\cot\gamma + \cot\alpha}\). Taking reciprocal on both sides, we get: \(\cot \beta = \frac{\cot\gamma + \cot\alpha}{2}\)

Step 3: Determine the progression

Now that the relation \(\cot \beta = \frac{\cot\gamma + \cot\alpha}{2}\) simplifies to a form where the middle term is the arithmetic mean of the other two terms, we can say that \(\cot \alpha, \cot \beta, \cot \gamma\) form an arithmetic progression.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the "common difference." For example, in the sequence 2, 4, 6, 8, the common difference is 2.
In this exercise, \( \cot \alpha, \cot \beta, \cot \gamma \) form an A.P. This means the difference between \( \cot \beta \) and \( \cot \alpha \) is the same as the difference between \( \cot \gamma \) and \( \cot \beta \):

• \( \cot \beta = \frac{\cot \alpha + \cot \gamma}{2} \)

This equation shows that the value of \( \cot \beta \) is exactly halfway between \( \cot \alpha \) and \( \cot \gamma \), confirming it forms an A.P.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variables. They are essential for simplifying expressions and solving trigonometric equations.
In this problem, the identity used was:

• \( \csc(\alpha + \gamma) = \frac{1}{\sin(\alpha + \gamma)} \)

This identity helps to rewrite \( \csc(\alpha + \gamma) \) in terms of \( \sin(\alpha + \gamma) \), making it easier to work with. Using identities to transform the original equation simplifies it significantly, leading to a clearer path to solve the exercise.

Sine and Cosine Rules

The sine and cosine rules are fundamental in working with angles and their trigonometric functions. In this context, they help express complex trigonometric relationships in a more understandable way.
The exercise used the sine addition formula:

• \( \sin(\alpha + \gamma) = \sin \alpha \cos \gamma + \cos \alpha \sin \gamma \)

This identity is pivotal in breaking down the given problem. It takes the complexities of combined angles and splits them into simpler parts involving individual sine and cosine terms. This breakdown allows us to rearrange and simplify expressions efficiently.

Cosecant Function

The cosecant function \( \csc \theta \) is the reciprocal of the sine function, defined as \( \csc \theta = \frac{1}{\sin \theta} \). It's crucial in scenarios where the sine of an angle is very small, and we need to focus on its reciprocal.
In this exercise, \( \csc(\alpha + \gamma) \) played a key role. By expressing it as \( \frac{1}{\sin(\alpha + \gamma)} \), it facilitated the simplification of the equation:

• \( \tan \beta = 2 \sin \alpha \cdot \sin \gamma \cdot \csc(\alpha + \gamma) \)
• This was rewritten to make it more workable: \( \tan \beta = \frac{2 \sin \alpha \cdot \sin \gamma}{\sin(\alpha + \gamma)} \)

Understanding the cosecant function allows for such transformations, aiding the simplification of complex trigonometric problems.