Problem 59 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\alpha, \beta\) be such that \(\pi<\alpha-\beta<3 \pi\). If \(\sin \alpha+\sin \beta=-\frac{21}{65}\) and \(\cos \alpha+\cos \beta=-\frac{27}{65}\), then the value of \(\cos \frac{\alpha-\beta}{2}\) is (A) \(-\frac{3}{\sqrt{130}}\) (B) \(\frac{3}{\sqrt{130}}\) (C) \(\frac{6}{65}\) (D) \(-\frac{6}{65}\)

Step-by-Step Solution

Answer

The value of \( \cos \frac{\alpha - \beta}{2} \) is \(-\frac{3}{\sqrt{130}}\).

Step 1: Use Trigonometric Identity

We begin by using the identities: \( \sin \alpha + \sin \beta = 2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) \) and \( \cos \alpha + \cos \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) \). Analyzing the given values, \( \sin \alpha + \sin \beta = -\frac{21}{65} \) and \( \cos \alpha + \cos \beta = -\frac{27}{65} \).

Step 2: Set Up Equation System

By setting the identities to the given values, we have two equations: 1. \( 2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) = -\frac{21}{65} \) 2. \( 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) = -\frac{27}{65} \).

Step 3: Solve for \( \cos \frac{\alpha - \beta}{2} \)

Divide equation 2 by equation 1: \[ \frac{2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)}{2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)} = \frac{-\frac{27}{65}}{-\frac{21}{65}} \] This simplifies to: \[ \cot \left( \frac{\alpha + \beta}{2} \right) = \frac{27}{21} = \frac{9}{7} \].

Step 4: Use Identity to Find \( \cos \frac{\alpha - \beta}{2} \) Value

Since \( \cot \frac{\alpha + \beta}{2} = \frac{9}{7} \), we need to further explore using trigonometric identities. With these ratios, we realize that our constraint \( \pi < \alpha - \beta < 3 \pi \) will assist in determining signs through the unit circle positioning. Calculating, \ \[ \cos \left( \frac{\alpha - \beta}{2} \right) = \pm \frac{3}{\sqrt{130}} \].

Step 5: Determine Correct Sign

Considering the range \( \pi < \alpha - \beta < 3 \pi \), implies that \( \frac{\pi}{2} < \frac{\alpha - \beta}{2} < \frac{3\pi}{2} \). In this region, the cosine function is negative, leading to \[ \cos \left( \frac{\alpha - \beta}{2} \right) = -\frac{3}{\sqrt{130}} \].

Key Concepts

Sine and Cosine Functions

One of the fundamental aspects of trigonometry involves the sine and cosine functions. These functions help us understand angles and lengths in right-angled triangles, and they extend to the unit circle, which allows us to represent these functions across all angles.

**Sine Function**
- It measures the vertical component or height of a point from the x-axis on the unit circle.- Given an angle \( \theta \), \( \sin\theta \) provides the y-coordinate of the point on the unit circle.

**Cosine Function**
- It measures the horizontal component or base of a point from the y-axis on the unit circle.- For \( \cos\theta \), it describes the x-coordinate of the point corresponding to angle \( \theta \) on the unit circle.

These functions are periodic, with sine and cosine cycling every \( 2\pi \) radians (360 degrees). Learning how to manipulate these functions through addition and multiplication provides tools for solving trigonometric equations.

Angle Difference Identities

Angle difference identities are powerful tools in trigonometry, which allow us to compute the sine, cosine, or tangent of an angle expressed as a difference between two other angles.

In our scenario, we utilized the following identities:

• \( \sin(\alpha) + \sin(\beta) = 2\sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right) \)
• \( \cos(\alpha) + \cos(\beta) = 2\cos \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right) \)

These identities break down the expressions of sums of sines and cosines into more manageable terms involving both angles' sums and differences. By using these identities, we can cleverly switch perspectives to solve complex trigonometric expressions more easily. In our problem, they allowed us to link known values of angle sums with unknowns concerning angles’ differences.

Trigonometric Equations

Trigonometric equations involve one or more trigonometric functions, and they can be used to find unknown angles or side lengths in a problem.

Solving these equations often requires the application of identities, formulas, and sometimes algebraic manipulation.

**Steps to Solve Trigonometric Equations**:

• Express the equation using known identities (as shown with angle difference identities).
• Simplify the equation by isolating the trigonometric functions.
• Solve for the variable by applying inverse trigonometric functions, adjusting according to the known constraints of the problem (using the range or domain of the angles).

In our exercise, starting with the given values for sine and cosine sums, we used these methodologies to find expressions for the difference of angles. This enabled us to solve for \( \cos \left(\frac{\alpha - \beta}{2}\right)\), where using constraints on the angle confirmed the correct sign. This showcases the importance of understanding the nature of trigonometric functions and identities when formulating solutions for real-world applications.