Problem 19 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(4 \cos ^{2} x \sin x-2 \sin ^{2} x=2 \sin x\)

Step-by-Step Solution

Answer

The solution to the equation are \(x = 0, \pi, \pi/6, 11\pi/6, \pi/2, 3\pi/2\)

Step 1: Rearranging the equation

Rearrange the equation such that all terms are on one side. So, we get \(4 \cos^{2} x \sin x -2 \sin^{2} x - 2 \sin x = 0\)

Step 2: Convert all terms into Sine

Express all the terms in terms of \( \sin x \). Use the Pythagorean identity \( \cos^{2} x = 1 - \sin^{2} x \). Substituting this into the equation gives \(4 (1 - \sin^{2} x) \sin x - 2 \sin^{2} x - 2 \sin x = 0 \). Simplifying results in \(4 \sin x - 4 \sin^{3} x - 2 \sin^{2} x - 2 \sin x = 0\).

Step 3: Rearranging the equation

Rearrange again to obtain the equation as \(-4 \sin^{3} x - 2 \sin^{2} x + 2 \sin x =0\).

Step 4: Factor out the equation

Factorization can be done by grouping common terms: \(-2 \sin x( 2 \sin^{2} x + \sin x -1 ) = 0\).

Step 5: Set equal to zero

Set each factor equal to zero and solve: \(2 \sin x = 0\) which gives \( \sin x = 0 \) and \(2 \sin^{2} x + \sin x - 1 = 0 \) can be solved using quadratic formula: \(\sin x=(-1± \sqrt{1+8})/4\). This gives: \(\sin x = -1/2\) and \(\sin x = 1\).

Step 6: Solve for \(x\)

Solving the equations \(\sin x = 0, \sin x = -1/2, \sin x = 1\). Results in \(x = 0, \pi, \pi/6, 11\pi/6, \pi/2, 3\pi/2\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry and is crucial in simplifying and solving trigonometric equations. It is given by \[\sin^2 x + \cos^2 x = 1\]This identity is derived from the Pythagorean theorem applied to the unit circle, where the hypotenuse is the radius (equal to 1). Understanding this identity allows us to express one of the functions, such as \(\cos^2 x\), in terms of the other function, \(\sin^2 x\). For example, if you know the value of \(\sin x\), you can easily find \(\cos x\) using:\[\cos^2 x = 1 - \sin^2 x\]In our original equation, we used this identity to replace \(\cos^2 x\) with \(1 - \sin^2 x\), which helps convert the equation into a form that contains only \(\sin x\), making it easier to solve.

• Remember: The Pythagorean Identity is always helpful when converting equations between sine and cosine.
• Use it to simplify expressions by reducing the number of different trigonometric terms.

Sine and Cosine functions

The sine and cosine functions are foundational in trigonometry. They describe the relationship between angles and side lengths in right triangles as well as coordinates on the unit circle. Sine, denoted as \(\sin x\), refers to the y-coordinate of a point on the unit circle, while cosine, or \(\cos x\), refers to the x-coordinate. When solving trigonometric equations, knowing how to express various components of the equation in terms of either sine or cosine can help streamline the solving process. In the original problem, converting all components to terms of \(\sin x\) reduced complexity.

• Tip: Always check if you can express an equation using a single trigonometric function (like we did with sine) to simplify the solving process.
• Sine and cosine values repeat every \(2\pi\), so solutions are often periodic and can have multiple values.

Approaching solutions with a clear understanding of these functions and their properties ensures confidence when tackling similar trigonometric equations.

Factorization of Trigonometric Functions

Factorization is a powerful mathematical tool used to solve equations by expressing them as a product of their factors. In trigonometric equations, factorization can simplify complex expressions and reveal simpler equations to solve. During the problem-solving process, once all terms were expressed in terms of \(\sin x\), factorization was applied:\[-2 \sin x( 2 \sin^2 x + \sin x - 1 ) = 0\]This step revealed two separate equations: \( \sin x = 0 \) and \( 2 \sin^2 x + \sin x - 1 = 0 \). Each factor equating to zero independently gives us distinct solutions. The quadratic factor was solved using the quadratic formula, a technique used when a simple factorization isn’t possible.

• Key Tip: Always look for common factors or use grouping to simplify trigonometric expressions.
• Once factorized, remember that each factor equating to zero provides potential solutions for the equation.

By effectively applying factorization techniques, complex trigonometric problems become more manageable, and solution pathways are made clearer.