Problem 24 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\alpha+\beta=\pi / 2\) and \(\beta+\gamma=\alpha\), then \(\alpha\) is (a) \(2(\tan \beta+\tan \gamma)\) (b) \((\tan \beta+\tan \gamma)\) (c) \((\tan \beta+2 \tan \gamma)\) (d) \((2 \tan \beta+\tan \gamma)\)

Step-by-Step Solution

Answer

The correct answer is (b) \(\alpha = (\tan \beta + \tan \gamma)\).

Step 1: Understand the given

We know that in a triangle, the sum of all angles is \( \pi \). From the given \(\alpha+\beta=\pi / 2\) and \(\beta+\gamma=\alpha\), we can deduce that \(\alpha + \gamma = \pi / 2\), since the sum of \(\alpha\), \(\beta\), and \(\gamma\) is \( \pi \) and \(\alpha + \beta = \pi / 2\). Therefore, \(\gamma\) can be written as \(\gamma = \pi / 2 - \alpha \).

Step 2: Substitute the value in given options

The goal is to prove one of four possible identities as \(\alpha\). So, let's substitute \(\gamma\) as \( \pi /2 - \alpha\) in all options and see which option simplifies to \(\alpha\). Option (a): \( \alpha = 2(\tan \beta + \tan (\pi / 2 - \alpha))\).Option (b): \( \alpha = (\tan \beta + \tan (\pi / 2 - \alpha))\). Option (c): \( \alpha = (\tan \beta + 2 \tan (\pi / 2 - \alpha))\). Option (d): \( \alpha = (2 \tan \beta + \tan (\pi / 2 - \alpha))\).

Step 3: Simplify options

We know that \(\tan ( \pi / 2 - \alpha) = \cot \alpha\). So, replacing \(\tan (\pi / 2 - \alpha)\) with \(\cot \alpha\) in all options, Option (a) becomes: \( \alpha = 2( \tan \beta + \cot \alpha)\). Option (b) simplifies to: \( \alpha = (\tan \beta + \cot \alpha)\).Option (c) simplifies to: \( \alpha = (\tan \beta + 2 \cot \alpha)\). Option (d) simplifies to: \( \alpha = (2 \tan \beta + \cot \alpha)\).

Step 4: Find the correct option

From the trigonometric idenitity \(\cot \alpha = 1 / \tan \alpha\), we can rewrite options a, b, c and d. After simplifying, only option b remains to be \( \alpha \). Therefore, the correct choice is option (b).

Key Concepts

Sum of Angles in a Triangle

The sum of angles in any triangle is a fundamental concept in geometry. It states that the interior angles of a triangle always add up to \( \pi \) radians, or 180 degrees. This fact helps us solve many problems involving triangles.

In the given problem, the conditions \( \alpha + \beta = \pi / 2 \) and \( \beta + \gamma = \alpha \) allow us to explore different relationships between these angles. By knowing that \( \alpha + \beta + \gamma = \pi \), we can deduce that \( \alpha + \gamma = \pi / 2 \).

From these relationships, we can express \( \gamma \) as \( \pi/2 - \alpha \), which helps us explore further simplifications and substitutions required in the problem. Understanding and applying the sum of angles rule is crucial for setting up the rest of the solution.

Tan and Cotangent Relations

Tangent and cotangent are important trigonometric functions that describe the ratio of sides in a right triangle. The tangent of an angle \( \theta \), denoted as \( \tan \theta \), is the ratio of the opposite side to the adjacent side. Cotangent, denoted as \( \cot \theta \), is the reciprocal of tangent, so \( \cot \theta = 1 / \tan \theta \).

In trigonometric identities, it's often useful to convert between tangent and cotangent. In our exercise, we take advantage of this by using the identity \( \tan( \pi / 2 - \alpha ) = \cot \alpha \) to simplify the expressions in the given options. This relationship helps us test which form of \( \alpha \) correctly satisfies the given conditions.

Understanding how these two functions relate is essential for simplifying trigonometric equations and solving problems that involve angle transformations.

Trigonometric Equation Simplification

Simplifying trigonometric equations involves using identities and algebraic manipulation to express complex terms in simpler forms. It allows us to identify equivalent expressions and verify solutions.

In the given exercise, after substituting \( \gamma = \pi / 2 - \alpha \), each option is rewritten using the identity \( \tan( \pi / 2 - \alpha) = \cot \alpha \). This simplification is essential to determine which option correctly represents \( \alpha \). Each expression is analyzed to see if it maintains equality.

Ultimately, option (b) simplifies to \( \alpha = (\tan \beta + \cot \alpha) \), meeting the conditions. Therefore, understanding and applying trigonometric identities to streamline equation solving is crucial in this problem.