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

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 \( \frac{\pi}{2} \), which is option (B).

Step 1: Understand the Problem

We need to determine the period of the function \( f(x) = \sin^4 x + \cos^4 x \), which involves analyzing how the function behaves over intervals and checking for repetition in values.

Step 2: Analyze Basic Functions

Understand that \( \sin x \) and \( \cos x \) both have a period of \( 2\pi \), meaning that \( \sin x = \sin(x + 2\pi) \) and \( \cos x = \cos(x + 2\pi) \). Thus, any function purely of \( \sin x \) or \( \cos x \) will have periodicity based on these base functions.

Step 3: Use Double Angle Formulas

Rewrite \( \sin^4 x + \cos^4 x \) using trigonometric identities. We use the identity \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 (2x) \), so:\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2 \left( \frac{1}{4} \sin^2 (2x) \right) \]Thus, \( f(x) = 1 - \frac{1}{2} \sin^2 (2x) \).

Step 4: Determine Periodicity

The function \( \sin(2x) \) has a period of \( \pi \) (half the period of \( 2\pi \) for \( \sin x \)), since a 2x factor compresses the period by a factor of 2. Therefore, \( \sin^2 (2x) \) has period \( \frac{\pi}{2} \) because squaring a sine function doubles the frequency.

Step 5: Conclude the Period of \( f(x) \)

As \( f(x) \) is expressed in terms of \( \sin^2 (2x) \) which itself has a period of \( \frac{\pi}{2} \), the function \( f(x) = 1 - \frac{1}{2} \sin^2 (2x) \) will also repeat every \( \frac{\pi}{2} \). So, the period of \( f(x) \) is \( \frac{\pi}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are crucial in simplifying expressions and solving equations. In our exercise, we encounter the expression

• \( \sin^4 x + \cos^4 x \).

To simplify this, we utilize the identity:

• \( \sin^2 x + \cos^2 x = 1 \).

Through manipulation,

• \( \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \).

This is a classic example of how trigonometric identities can convert complex expressions into simpler forms. This specific expression eventually simplifies our target function to

• \( f(x) = 1 - \frac{1}{2} \sin^2(2x) \).

Knowing these identities makes analyzing the periodic nature of functions like this much simpler.

Function Analysis

Function analysis involves understanding the behavior, characteristics, and properties of a function. Analyzing a function like

• \( f(x) = \sin^4 x + \cos^4 x \)

starts with recalling that both basic sine and cosine functions individually have a period of

• \( 2\pi \).

The next step is transforming the given function to observe its periodicity better. By using

• double angle and square identities,

we rewrite the function as

• \( f(x) = 1 - \frac{1}{2} \sin^2 (2x) \).

This transformation shows the effect of combining sine and cosine powers and analyzing their symmetry and repetition over intervals. Understanding these transformations can reveal more about the function’s peaks, troughs, and repeating intervals, greatly aiding in determining its period.

Double Angle Formulas

Double angle formulas are key tools in trigonometry for expressing trigonometric functions at double the angle in terms of single angles. A common double angle formula for sine is:

• \( \sin(2x) = 2 \sin x \cos x \).

For squaring

• \( \sin(2x) \),

we reach

• \( \sin^2(2x) \),

which greatly impacts periodicity. In our function,

• \( f(x) = 1 - \frac{1}{2} \sin^2 (2x) \),

we see the function's periodicity is directly influenced by \( \sin^2(2x) \).The double angle formula compresses the basic sine period

• \( 2\pi \)

into

• \( \pi \),

while further manipulations, like squaring, halve it again to

• \( \frac{\pi}{2} \).

Using double angle formulas simplifies both the calculation and understanding of the repeats within trigonometric functions.