Problem 25 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha\) and \(\beta\) be two distinct values of \(\theta\) lying between 0 and \(2 \pi\), satisfying the equation \(3 \cos \theta+4 \sin \theta=2\), then the value of \(\sin (\alpha+\beta)\) is (A) \(\frac{12}{25}\) (B) \(\frac{24}{25}\) (C) \(\frac{13}{25}\) (D) none of these

Step-by-Step Solution

Answer

The value of \(\sin(\alpha + \beta)\) is 0.

Step 1: Introduce the Problem using an Identity

Given the equation \(3 \cos \theta + 4 \sin \theta = 2\), where \(\theta = \alpha\) or \(\theta = \beta\), we can use the identity \(R \cos(\theta + \phi) = a \cos \theta + b \sin \theta\) with \(R = \sqrt{a^2 + b^2}\).

Step 2: Calculate R and Adjust the Equation

Let's apply the values from the given equation. Here, \(a = 3\) and \(b = 4\). Therefore, \[ R = \sqrt{3^2 + 4^2} = \sqrt{9+16} = 5 \]. Thus, the equation becomes \[ 5 \cos(\theta + \phi) = 2 \], which gives us \(\cos(\theta + \phi) = \frac{2}{5}\).

Step 3: Express in Terms of Cosine and Sine

When \(\theta = \alpha\), \(\cos(\alpha + \phi) = \frac{2}{5}\). Similarly, when \(\theta = \beta\), \(\cos(\beta + \phi) = \frac{2}{5}\) because they satisfy the same equation. Since \(\alpha\) and \(\beta\) are distinct angles, they are symmetric about the origin in terms of the cosine value, giving \(\alpha + \beta = 2k\pi\) or \(2k\pi + \pi\).

Step 4: Determine \(\sin(\alpha + \beta)\)

Using the formula \(\sin(\alpha + \beta) = \sin(\alpha + \phi + \beta + \phi - 2\phi) = \sin(2\phi - 2\phi) = \sin(0)\), we have \(\sin(0) = 0\) since the angles in symmetry about the \(x\)-axis or full rotation result in zero sine value.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving equations involving trigonometric functions like sine, cosine, and tangent. In this problem, the identity used is an expression of a linear combination of sine and cosine functions in the form:

• \( R \cos(\theta + \phi) = a \cos \theta + b \sin \theta \)

Here, the constants \(a\) and \(b\) are coefficients of cosine and sine, respectively, and \(R\) is computed as \(\sqrt{a^2 + b^2}\).Using this identity helps convert the problem into a single cosine term with a phase shift \(\phi\), making it easier to analyze. This identity also allows us to identify angles that lead to trigonometric equations being satisfied, as shown in the problem's transformation.

Cosine Addition Formula

The cosine addition formula, \( \cos(A + B) = \cos A \cos B - \sin A \sin B \), expresses the cosine of the sum of two angles in terms of the sines and cosines of the individual angles. This formula is helpful in breaking down compound angles and helps establish relationships between angles in a trigonometric equation.In the given problem, this formula aids in expressing the trigonometric equation using simplified terms. The conversion to \( \cos(\theta + \phi) = \frac{2}{5} \) relies on recognizing that distinct angles \(\alpha\) and \(\beta\) correspond to the same equation form. By determining \(R\) and expressing as a cosine equation, the addition formula facilities the understanding that \(\alpha + \beta\) can produce a sum reflecting symmetry or periodic properties of cosine.

Sine Properties

The properties of the sine function are cornerstones in trigonometry, often revealed through symmetrical and periodic behaviors. The sine function, which gives values between -1 and 1, is crucial in comprehending waveforms and oscillations in mathematics.In this exercise, we exploit the symmetry of sine for angles \(\alpha\) and \(\beta\). Since \(\cos(\theta + \phi) = \frac{2}{5}\) is the same for both \(\alpha\) and \(\beta\), it suggests that they are symmetrically positioned about a midpoint due to cosine's even function nature. This symmetry implies that when you sum these angles, their sine will yield:

• \(\sin(\alpha + \beta) = 0\)

The zero result arises because the angles reflect across the axis, fitting trigonometric periodic rules and wave-like repetition. Recognizing these properties enables simplification especially when solving equations by establishing where sine equals zero.