Problem 11 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The general solution of \(\cos ^{5} x-\sin ^{5} x-1=0\) is (a) \(n \pi\) (b) \(2 n \pi\) (c) \(n \pi+\frac{\pi}{2}\) (d) \(2 n \pi+\frac{\pi}{2}\)

Step-by-Step Solution

Answer

The correct solution is option (c) \(x = n\pi + \frac{\pi}{2}\), where n is any integer.

Step 1: Generate a simplified equation

Rewrite the equation as \(\cos ^{2} x \cdot \cos ^{3} x - \sin ^{2} x \cdot \sin ^{3} x - 1 = 0\). Apply the identity \(\cos ^{2} x + \sin ^{2} x = 1\). Split the equation to look like \((\cos^2 x) (\cos^3 x - \sin^3 x) - 1 = 0\). Again, use the identity to rewrite \(\cos^3 x - \sin^3 x\) as \((\cos x + \sin x) (\cos^2 x - \sin x \cos x + \sin^2 x)\). Now the equation simplifies to \((\cos^2 x) (\cos x + \sin x) (\cos^2 x - \sin x \cos x + \sin^2 x) - 1= 0\) .

Step 2: Use identities to solve the trigonometric equation

To find the solutions for x, one of each of the multipliers in the equation needs to be equal to zero: either \(\cos^2 x = 0\), \(\cos x + \sin x = 0\), or \(\cos^2 x - \sin x \cos x + \sin^2 x = 1\). Solving each equation, you'll find the solutions for \(\cos^2 x = 0\) are \(x = \frac{\pi}{2} + n\pi\) (for integers n), and the solutions for \(\cos x + \sin x = 0\) are \(x = \frac{3\pi}{4} + n\pi\). The third equation always holds true, hence doesn't contribute extra solutions.

Step 3: Comparison with the provided solutions

Upon comparing the solutions with the options, it turns out that the general form is given by option (c): \(x = n\pi + \frac{\pi}{2}\), where n is any integer.

Key Concepts

General Solution

When dealing with trigonometric equations, such as \(\cos^{5}x - \sin^{5}x - 1 = 0\), a goal is often to find a general solution. This general solution is a formula that represents all possible solutions of the equation. In this case, it was determined that the solutions are of the form \(x = n\pi + \frac{\pi}{2}\), with \(n\) being any integer.

• The concept of a general solution allows us to capture multiple solutions at once using a single formula.
• This approach is particularly useful because trigonometric functions are periodic, meaning they repeat values over regular intervals.

Thus, finding a general solution can simplify the process of solving equations and give us a comprehensive understanding of all possible solutions.

Trigonometric Identities

Trigonometric identities are formulas that help simplify and solve trigonometric equations. For instance, one fundamental identity is \(\cos^2 x + \sin^2 x = 1\). This identity was used in simplifying the given equation, allowing us to express and rearrange terms.

• By using identities, complex expressions can break down into simpler, more manageable pieces.
• For example, the expression \(\cos^3 x - \sin^3 x\) can be rewritten using trigonometric identities into \((\cos x + \sin x)(\cos^2 x - \sin x \cos x + \sin^2 x)\).

These identities are powerful tools not just for solving equations but also for proving more advanced trigonometric theorems.

Cosine and Sine Functions

The cosine and sine functions are at the core of many trigonometric equations. Each function has its own properties, such as range and domain, and their periodic nature.

• Cosine and sine repeat every \(2\pi\) radians. Thus, their periodic nature provides an infinite number of solutions for equations like \(\cos^{2}x = 0\), resulting in the form \(x = \frac{\pi}{2} + n\pi\).
• When summed, as in \(\cos x + \sin x\), they create harmonic relationships that can further simplify solving equations.

Understanding these properties is crucial for effectively solving equations that include trigonometric functions. Recognizing patterns and relationships in these functions helps to predict solutions more accurately and quickly.