Problem 12 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following trigonometric equations: \(\sin ^{4} x+\sin ^{4}\left(x+\frac{\pi}{4}\right)=\frac{1}{4}\)

Step-by-Step Solution

Answer

The solution involves applying the trigonometric identity \(\sin(a+b)= \sin a \cos b + \cos a \sin b\), followed by several simplifications and solving a quartic equation

Step 1: Apply the trigonometric identity

Start from the given equation and rewrite \(\sin ^{4}(x+\frac{\pi}{4})\) using the trigonometric identity:\(\sin ^{4} x + (\sin(x) \cos(\frac{\pi}{4}) + \cos(x) \sin(\frac{\pi}{4}))^{4} = \frac{1}{4}\)

Step 2: Simplify the equation

Note that \(\sin(\frac{\pi}{4})\) and \(\cos(\frac{\pi}{4})\), both equals to \(\frac{1}{\sqrt{2}}\), so the equation can be further simplified to:\(\sin ^{4} x + \frac{(\sin(x) + \cos(x))^{4}}{4} = \frac{1}{4}\)

Step 3: Subtract \(\sin ^{4} x\) from both sides

\((\sin(x) + \cos(x))^{4} = \frac{1}{4} - \sin ^{4} x\)

Step 4: Apply the Pythagorean identity

We can rearrange the last equation in terms of the Pythagorean identity, \(\sin^2 x + \cos ^2 x = 1\):\((\sin(x) + \cos(x))^{4} = \frac{1}{4} - (1 - \cos^2 x)^2\)This simplifies to:\((\sin(x) + \cos(x))^{4} = \frac{1}{4} - (1 - 2\cos^2 x + cos^4 x)\)

Step 5: Solve for \(x\)

The above equation simplifies further to a quartic (4th order) equation in \(\cos x\b), which can be solved for \(x\)

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions, such as sine, cosine, and tangent, which hold true for all values within the domains of the functions. To resolve complex trigonometric equations, like in the original exercise, identifying and applying these identities correctly is crucial.

In the problem, a trigonometric identity is applied to rewrite the expression \( \sin^4\left(x+\frac{\pi}{4}\right) \) into a more workable form. The identity leverages the angle sum formula: \( \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \). By understanding these principles, one can transform and simplify expressions, often a first step in solving trigonometric equations.

The Pythagorean Identity

The Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \), is a fundamental concept derived from the Pythagorean theorem related to right-angled triangles. It connects the square of the sine and cosine of an angle to form the constant 1.

In our step-by-step solution, Step 4 effectively uses this identity to transform the quartic equation. By recognizing \( \sin^4 x \) as \((1 - \cos^2 x)^2\), the problem is greatly simplified. This showcases the power of the Pythagorean identity in changing the form of trigonometric equations, moving us toward a resolution. The use of these identities are indispensable tools in a mathematician's toolkit when dealing with trigonometric equations.

Quartic Equations

A quartic equation is a type of polynomial equation of the fourth degree, meaning it has the highest power of 4. The general form is given by \(ax^4 + bx^3 + cx^2 + dx + e = 0\). While more complex than quadratic or cubic equations, some quartic equations can still be solved exactly by a combination of algebraic manipulation and known formulas.

In the provided exercise, the solution eventually simplifies to a quartic equation in terms of \(\cos x\). Solving quartic equations often requires recognizing patterns and applying strategic algebraic techniques, as well as possibly factoring or using specialized methods like Ferrari's solution when relevant.