Problem 36 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The number of solution of the equation \(1+\sin x \cdot \sin ^{2} \frac{x}{2}=0\) in \([-\pi, \pi]\) is (a) 0 (b) 1 (c) 2 (d) 3

Step-by-Step Solution

Answer

After simplifying and solving the equation, it is found that the equation has 2 solutions in the given range. Therefore, the correct answer is option (c) 2.

Step 1: Simplify the equation

The given equation is \(1+ \sin x \cdot \sin^2 \frac{x}{2} = 0\). We can multiply out the right hand side of the equation to get \(\sin x \cdot \sin^2 \frac{x}{2} = -1\)

Step 2: Use double-angle formula

The double-angle formula for sine is \(\sin 2x = 2 \sin x \cos x\). Applying this to our equation we get \(2 \sin \frac{x}{2} \cos \frac{x}{2} = -1\)

Step 3: Solve for x

Solve the equation \(2 \sin \frac{x}{2} \cos \frac{x}{2} = -1\) for \(x\). The solutions which satisfy \(- \pi \leq x \leq \pi\) need to be identified and counted. These would constitute the roots of the original equations.

Key Concepts

Double-Angle Formulas

Understanding double-angle formulas is crucial for simplifying and solving trigonometric equations. These formulas are derived from the sum formulas of trigonometric functions and provide a way to express trigonometric functions of angles like \(2x\) in terms of \(x\). For instance, the double-angle formula for sine, which is essential in the given exercise, is \(\sin 2x = 2 \sin x \cos x\).

Similarly, we have double-angle formulas for cosine and tangent:

• \( \cos 2x = \cos^2 x - \sin^2 x\) which can also be written as \(1 - 2\sin^2 x\) or \(2\cos^2 x - 1\)
• \(\tan 2x = \frac{2\tan x}{1 - \tan^2 x}\)

These identities are fundamental when dealing with trigonometric equations, as they allow you to simplify expressions with multiple angles and solve for the variable involved.

Trigonometric Identities

Trigonometric identities are equations that relate trigonometric functions to one another and are true for all values within their domains. They're indispensable for transforming and simplifying trigonometric expressions. Some basic identities include the Pythagorean identities, such as \( \sin^2 x + \cos^2 x = 1 \) and its derived forms \( 1 + \tan^2 x = \sec^2 x \) and \( \cot^2 x + 1 = \csc^2 x \).

Additionally, there are ratio identities like \( \sin x = \frac{opposite}{hypotenuse} \) and reciprocal identities such as \( \sec x = \frac{1}{\cos x} \). Using these identities effectively in solving trigonometric equations simplifies complex expressions and makes it possible to solve for unknown angles and side lengths in right triangles.

Solutions of Trigonometric Equations

Solving trigonometric equations involves finding all the angle measures that satisfy the equation. The approach often requires utilizing trigonometric identities to rewrite the equation in a more solvable form. For example, in the given exercise, we utilize a double-angle formula to simplify the equation before solving for \(x\).

It's important to consider the domain in which we're searching for solutions. For instance, the interval \([-\pi, \pi]\) refers to all angles measuring between -180 and 180 degrees. When solving for \(x\), it's necessary to find only the solutions that lie within the given domain, ignoring extraneous or out-of-bound solutions. To present the solutions accurately, one can use the unit circle or reference angles to determine where the solutions lie within the specified domain. In conclusion, solving for trigonometric equations often yields multiple angles, and attention must be paid to the domain and periodicity of the trigonometric functions involved.