Problem 18 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(4 \sin ^{4} x+\cos ^{4} x=1\)

Step-by-Step Solution

Answer

The solutions for the given equation are \(x = n\pi\), \(x = \frac{\pi}{2} + n\pi\), and \(x = -\frac{\pi}{2} + n\pi\), where \(n\) is an integer.

Step 1: Substitute \(\sin^2 x\)

We can express \(\sin^4 x = (\sin^2 x)^2\), so rewrite the equation as \(4(\sin^2 x)^2 + \cos^4 x = 1\)

Step 2: Substitute \(\cos^2 x\) in terms of \(\sin^2 x\)

The Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) can be rewritten as \(\cos^2 x = 1 - \sin^2 x\). Substitute this into our equation to get \(4(\sin^2 x)^2 + (1 - \sin^2 x)^2 = 1\)

Step 3: Solve the Quadratic Equation

At this point, the equation becomes a quadratic equation in terms of \(\sin^2 x\), which we solve for \(\sin^2 x\) giving possible values of \( \sin^2 x = 1, 0 \)

Step 4: Determine Possible Values for x

If \(\sin^2 x = 1\) then \(\sin x = \pm 1\) which has solutions \(x = \frac{\pi}{2} + n\pi\), and \(x = -\frac{\pi}{2} + n\pi\). If \(\sin^2 x = 0\) then \(\sin x = 0\) which gives solutions \(x = n\pi\), where \(n\) is an integer.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental relations in trigonometry. It is expressed as \(\sin^2 x + \cos^2 x = 1\). This identity comes from the geometric nature of a right triangle in the coordinate plane. Imagine a unit circle, which is a circle with radius 1 centered at the origin of a coordinate system. The x-coordinate of any point on this circle can be expressed as \(\cos x\) and the y-coordinate as \(\sin x\).
Because the radius is 1, the relationship between the x and y-coordinates follows the Pythagorean theorem: the sum of the squares of the x and y coordinates equals the square of the radius (1). This is why the Pythagorean identity holds true. It is very useful for simplifying trigonometric expressions by letting us substitute \(\cos^2 x\) with \(1 - \sin^2 x\), or vice versa.
This identity not only aids in equation solving, as seen in the original problem, but is also crucial for integrating and differentiating trigonometric functions as part of calculus.

Trigonometric equations

Trigonometric equations involve trigonometric functions and are solved for angles that satisfy the given conditions. These equations are crucial for analyzing periodic phenomena in various fields, including physics and engineering.
When solving trigonometric equations, such as \(4 \sin^4 x + \cos^4 x = 1\), the aim is to find the specific values of \(x\) that will satisfy the equation. To do this, you might need to use identities like the Pythagorean identity to transform the equation into a simpler form. By rewriting the equation to contain only one trigonometric function, it becomes more manageable.
It is common to encounter situations where breaking the problem down by using substitutions helps. In this problem, replacing \(\cos^2 x\) with \(1 - \sin^2 x\) using the Pythagorean identity allowed us to reduce the equation to a form that could be handled as a quadratic equation in \(\sin^2 x\). This is a standard technique in handling trigonometric equations.

Quadratic equations

A quadratic equation is a type of polynomial equation of the second degree, typically written in the form \(ax^2 + bx + c = 0\). Solving quadratic equations often involves finding the values of the unknown variable (in this case, \(\sin^2 x\)) that satisfy the equation. Quadratic equations are prevalent in both algebra and trigonometry.
To solve a quadratic equation, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), factor the equation if possible, or complete the square. For the equation in our exercise, after transforming the original trigonometric equation, we looked for real solutions for \(\sin^2 x\), and we found two: 1 and 0.
These values of \(\sin^2 x\) were then used to find potential angles \(x\) in radians. This process demonstrates how transforming a problem using identities can lead to a solvable formation using well-known algebraic methods.