Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

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 is \(\frac{3}{\sqrt{130}}\).

Step 1: Use Sum-to-Product Identities

Convert the sum of sines and cosines into products. The sum-to-product identities are: \(\sin \alpha + \sin \beta = 2 \sin \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right)\) and \(\cos \alpha + \cos \beta = 2 \cos \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right)\).

Step 2: Form Equations Using Given Values

Substitute the given values into the identities: \( 2 \sin \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right) = -\frac{21}{65}\) and \( 2 \cos \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right) = -\frac{27}{65}\).

Step 3: Isolate \(\cos \left( \frac{\alpha-\beta}{2} \right)\)

Let \(x = \cos \left( \frac{\alpha-\beta}{2} \right)\). From our equations, we have \( x \sin \left( \frac{\alpha+\beta}{2} \right) = -\frac{21}{130}\) and \(x \cos \left( \frac{\alpha+\beta}{2} \right) = -\frac{27}{130}\).

Step 4: Use Pythagorean Identity

Since there is a constraint that involves a sum of squares (\(\sin^2 + \cos^2 = 1\)), divide \( x^2 \left( \sin^2 \left( \frac{\alpha+\beta}{2} \right) + \cos^2 \left( \frac{\alpha+\beta}{2} \right) \right) = \left(\frac{21}{130}\right)^2 + \left(\frac{27}{130}\right)^2\). Simplifying gives \(x^2 = \frac{30}{130} = \frac{3}{13}\).

Step 5: Solve for \(x\)

Calculate \(x = \pm \sqrt{\frac{3}{13}}\). Since \(\pi < \alpha - \beta < 3\pi\), \(\alpha - \beta\) is an angle between \(\pi\) and \(2\pi\), so \(\cos \left( \frac{\alpha-\beta}{2} \right)\) should be positive for an angle less than but not exceeding \(\pi\). Therefore, \(x = \frac{3}{\sqrt{130}}\).

Key Concepts

Sum-to-Product Identities

In trigonometry, understanding identities is essential for simplifying expressions and solving equations. Sum-to-Product Identities are particularly helpful when working with sums of sine and cosine. These identities allow us to convert sums into products, which can simplify the process of solving trigonometric equations. For example, the identity for sine is:

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

For cosine, it looks like this:

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

When applied to problems like the given exercise, these identities help us express the sums of sines and cosines in terms of products that involve the half-sum and half-difference of angles. This offers a path to find other unknowns, like \( \cos \frac{\alpha-\beta}{2} \), by forming new equations based on the rearranged identities.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions such as sine, cosine, and tangent. These equations often appear in problems relating to periodic phenomena or rotations. Solving trigonometric equations involves various strategies, including algebraic manipulation, using identities, and understanding the properties of trig functions.

In this exercise, once we substitute the values into the sum-to-product identities, we form two separate equations:

• \( 2 \sin \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right) = -\frac{21}{65} \)
• \( 2 \cos \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right) = -\frac{27}{65} \)

These equations can be solved by isolating \( \cos \left( \frac{\alpha-\beta}{2} \right) \). It involves some algebraic manipulation where the properties of sine and cosine will be crucial. The ability to transform these equations provides insight into both the nature of the trigonometric functions and the specific values we seek.

Pythagorean Identity

The Pythagorean Identity is a fundamental identity in trigonometry that relates the square of sine and cosine of an angle. It states:

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

This identity is essential when dealing with trigonometric equations involving squares. It allows us to transform and simplify expressions by substituting concepts freely between \( \sin^2 \) and \( \cos^2 \).

In the current problem, the Pythagorean Identity is utilized after isolating \( x = \cos \left( \frac{\alpha-\beta}{2} \right) \). We reached the equation:

• \( x^2 \left( \sin^2 \left( \frac{\alpha+\beta}{2} \right) + \cos^2 \left( \frac{\alpha+\beta}{2} \right) \right) = \left( \frac{21}{130} \right)^2 + \left( \frac{27}{130} \right)^2 \)

Using \( \sin^2 + \cos^2 = 1 \), we simplify it to \( x^2 = \frac{3}{13} \), making it easier to solve for \( x \), providing a clear path to finding \( \cos \frac{\alpha-\beta}{2} \). Understanding and applying the Pythagorean Identity effectively can literally reduce the complexity and number of steps required in solving various trigonometric problems.