Problem 152 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The period of the function \(f(x)=\sin ^{4} x+\cos ^{4} x\) is: (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(2 \pi\) (D) None of these

Step-by-Step Solution

Answer

The period is \( \frac{\pi}{2} \).

Step 1: Express Power Formulas

To simplify the given function, we use trigonometric identities. We start by expressing powers of trigonometric functions: \( \sin^4 x = (\sin^2 x)^2 \) and \( \cos^4 x = (\cos^2 x)^2 \).

Step 2: Apply Double Angle Identity

Recall that \( \sin^2 x = \frac{1 - \cos 2x}{2} \) and \( \cos^2 x = \frac{1 + \cos 2x}{2} \). We apply these identities: \( \sin^4 x = \left(\frac{1 - \cos 2x}{2}\right)^2 \) and \( \cos^4 x = \left(\frac{1 + \cos 2x}{2}\right)^2 \).

Step 3: Simplify the Expression

Now add these expressions: \( f(x) = \left(\frac{1 - \cos 2x}{2}\right)^2 + \left(\frac{1 + \cos 2x}{2}\right)^2 \). Expand and combine terms.

Step 4: Expand Squared Terms

We expand \( \left(\frac{1 - \cos 2x}{2}\right)^2 = \frac{1 - 2\cos 2x + \cos^2 2x}{4} \) and \( \left(\frac{1 + \cos 2x}{2}\right)^2 = \frac{1 + 2\cos 2x + \cos^2 2x}{4} \).

Step 5: Add the Expanded Expressions

Adding the expanded expressions results in \( f(x) = \frac{2 + 2\cos^2 2x}{4} = \frac{1}{2} + \frac{\cos^2 2x}{2} \).

Step 6: Replace \( \cos^2 2x \) with Identity

Use \( \cos^2 2x = \frac{1 + \cos 4x}{2} \). Substitute this identity back: \( f(x) = \frac{1}{2} + \frac{1 + \cos 4x}{4} = \frac{3}{4} + \frac{\cos 4x}{4} \).

Step 7: Determine the Period of the Function

The term \( \cos 4x \) has a period of \( \frac{2\pi}{4} = \frac{\pi}{2} \). Therefore, \( f(x) = \frac{3}{4} + \frac{\cos 4x}{4} \) also has a period of \( \frac{\pi}{2} \).

Key Concepts

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. A classic example is the sine or cosine function, which both have the basic period of \(2\pi\). These functions are fundamental in trigonometry and many real-world applications, such as sound waves and alternating current circuits.

The key to understanding periodic functions is to recognize how their periods are manipulated. There is a simple relationship: if the function is altered by multiplying its variable by a constant, the period is divided by that constant. For example:

• The function \(\sin x\) has a period of \(2\pi\).
• \(\sin(2x)\) has a period of \(\pi\) because the variable \(x\) is multiplied by 2.
• The period becomes \(\frac{2\pi}{n}\) when the frequency is multiplied by \(n\).

In the given exercise, the function \(f(x) = \sin^4 x + \cos^4 x\) was transformed into an expression containing \(\cos 4x\). The periodicity of the whole function is governed by \(\cos 4x\), which modifies the original period of \(2\pi\) to \(\frac{\pi}{2}\).

Thus, it's important to remember how transformations affect periods. This knowledge helps solve problems involving trigonometric functions quickly.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions, like sine and cosine, that are true for every value of the involved variables. These identities are foundational tools for simplifying and solving trigonometric problems.

In the exercise, we used some basic trigonometric identities to simplify \(\sin^4 x + \cos^4 x\). The identities applied include:

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

These identities are derived from the double angle identities and help express powers of trigonometric functions in terms of angles that are easier to work with. They support not just algebraic manipulation but also provide better insights into the properties of trigonometric functions.

Additionally, there are many more trigonometric identities, including the Pythagorean identities \(\sin^2 x + \cos^2 x = 1\) and angle sum formulas, which are invaluable in mathematics. Using these identities can turn complex expressions into simpler forms, making it easier to find periods, solve equations, and understand trigonometric behavior in various contexts.

For students, familiarity with these identities is key to mastering trigonometry and efficiently solving related mathematical problems.

Double Angle Identities

Double angle identities are a special set of trigonometric identities that relate the trigonometric functions of double angles to the functions of single angles. They are particularly useful when dealing with expressions involving squared trigonometric terms.

One of the fundamental double angle identities used in the exercise was:

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

These identities allow you to express squares of sine and cosine as linear expressions, thereby simplifying the calculations and algebra involved in the problem.

The use of double angle identities is crucial in reducing quadratic trigonometric expressions to forms that are easier to integrate or differentiate. They are also essential when determining the amplitudes, phases, and periods of trigonometric functions.

In solving the period of \(f(x) = \sin^4 x + \cos^4 x\), we relied on double angle identities to effectively reduce the expression to a simpler form, leading to the correct identification of the function's period as \(\frac{\pi}{2}\). Double angle identities, thus, play a vital role in both simplifying expressions and understanding the dynamic changes in periodic functions.