Problem 106 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin x+2 \cos x=\cos 2 x-\sin 2 x $$

Step-by-Step Solution

Answer

This exercise does not have an exact algebraic solution. Therefore, there is no short and final answer which can be computed directly. Instead a numerical approximation method will need to be applied to approximate the solutions.

Step 1: Rewriting the equation

By using the identity for cosine of the double angle, \(\cos 2x = \cos^2x - \sin^2x \), the equation can be rewritten as \(\sin x + 2\cos x = \cos^2x -\sin^2x\).

Step 2: Grouping the terms

Regroup the terms to gather all sine terms and cosine terms on one side each, \(\sin^2x - \sin x + \cos^2x - 2\cos x = 0.\)

Step 3: Use the Pythagorean identity

Substitute \(\sin^2x =\ 1 - \cos ^2x\) using Pythagorean identity, \(1 - \cos ^2x - \sin x - \cos ^2x - 2\cos x = 0\). It simplifies to \(1- \sin x -3\cos ^2x - 2\cos x = 0\).

Step 4: Substitute \(y = \cos x\)

We can make a substitution to simplify the equation. Substituting \(y = \cos x\) gives \(1 - 3y^2 - 2y - \sqrt{1-y^2} = 0\). We now need to solve this quadratic equation with an additional square root term.

Step 5: Solving the equation

From the equation in Step 4, the roots cannot be found directly due to the square root term. This exercise does not have a direct solution for x that can be computed algebraically, but solutions can be approximated numerically.

Key Concepts

Double Angle Identity

Understanding the double angle identity is crucial when solving trigonometric equations like \( \text{sin} x + 2 \text{cos} x = \text{cos} 2x - \text{sin} 2x \). Double angle identities refer to the trigonometric identities where the angle in the trigonometric function is doubled. The most commonly used double angle identities involve sine, cosine, and tangent.

For instance, the cosine double angle identity can be expressed in three ways:

• \( \text{cos} 2x = \text{cos}^2 x - \text{sin}^2 x \)
• \( \text{cos} 2x = 2 \text{cos}^2 x - 1 \)
• \( \text{cos} 2x = 1 - 2 \text{sin}^2 x \)

In the initial step of the solution, the first variation of the cosine double angle identity is applied, which breaks down \( \text{cos} 2x \) into \( \text{cos}^2 x - \text{sin}^2 x \). This allows for the trigonometric equation to be rearranged more simply.

Pythagorean Identity

The Pythagorean identity is another bedrock in trigonometry, which helps simplify equations by connecting sine and cosine to the unit circle. The most fundamental Pythagorean identity is \( \text{sin}^2 x + \text{cos}^2 x = 1 \). This relationship enables us to express \( \text{sin}^2 x \) as \( 1 - \text{cos}^2 x \) and vice versa.

In the provided solution, the Pythagorean identity is used to substitute \( \text{sin}^2 x \) with \( 1 - \text{cos}^2 x \), which transforms the trigonometric equation into a more manageable form. This substitution is a strategic move that reduces the complexity of the equation, and is particularly helpful when equations contain both sine and cosine terms.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify equations by replacing trigonometric functions with a single variable. This approach is especially useful when dealing with quadratic forms in trigonometry. In the context of solving the given equation, \( y = \text{cos} x \) is employed.

By substituting \( y \) for \( \text{cos} x \), the equation becomes \( 1 - 3y^2 - 2y - \text{sqrt}1-y^2} = 0 \), which is similar to a quadratic equation with an additional radical. Although the step by step solution suggests that the equation does not lead to an algebraic solution for \( x \), the substitution helps in simplifying the expression and is a gateway to numerical methods for approximation. Trigonometric substitution is an invaluable tool when dealing with complex trigonometric equations, simplifying them to a form that may be more amenable to various solving techniques.