Problem 71 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A=\sin ^{2} x+\cos ^{4} x\) then, for all real values of \(x\), (A) \(\frac{13}{16} \leq A \leq 1\) (B) \(1 \leq A \leq 2\) (C) \(\frac{3}{4} \leq A \leq \frac{13}{16}\) (D) \(\frac{3}{4} \leq A \leq 1\)

Step-by-Step Solution

Answer

(D) \( \frac{3}{4} \leq A \leq 1 \)

Step 1: Understanding the Expression

The given expression is \( A = \sin^2 x + \cos^4 x \). We want to find the range of \( A \) for all real values of \( x \).

Step 2: Use Trigonometric Identities

Recall the identity \( \sin^2 x + \cos^2 x = 1 \). From this, we can express \( \cos^2 x = 1 - \sin^2 x \). Therefore, \( \cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2 \).

Step 3: Substitute and Simplify

Substitute \( \cos^4 x = (1 - \sin^2 x)^2 \) into the expression for \( A \): \[A = \sin^2 x + (1 - \sin^2 x)^2.\] Now expand \((1 - \sin^2 x)^2\): \(1 - 2\sin^2 x + \sin^4 x \). So, \( A = \sin^2 x + 1 - 2\sin^2 x + \sin^4 x = 1 - \sin^2 x + \sin^4 x \).

Step 4: Analyze the Expression

Let \( y = \sin^2 x \). Then \( 0 \leq y \leq 1 \) for all real \( x \). Substitute to get \( A = 1 - y + y^2 \).

Step 5: Complete the Square

Rewrite \( A = y^2 - y + 1 \). Completing the square gives \( A = (y - \frac{1}{2})^2 + \frac{3}{4} \). Since \((y - \frac{1}{2})^2 \geq 0\), we conclude \( \frac{3}{4} \leq A \).

Step 6: Determine Upper Bound

For the upper bound, since the maximum value of the square is when \((y - \frac{1}{2})^2 = 0\), \( A \) reaches its maximum value of 1. Thus, \( A \leq 1 \).

Step 7: Conclusion

The range of \( A \) is \( \frac{3}{4} \leq A \leq 1 \).

Key Concepts

Sine and Cosine Functions

Understanding sine and cosine functions is essential in resolving trigonometric equations. These functions relate to the angles and sides of a right triangle. Specifically,

• Sine (\( \sin \)) of an angle is the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)) is the ratio of the adjacent side to the hypotenuse.

These functions are periodic, meaning they repeat their values in a regular pattern as the angle increases. A key property is that the square of \( \sin x \) plus the square of \( \cos x \) is always one: \( \sin^2 x + \cos^2 x = 1 \).

In the given exercise, the functions’ periodic nature ensures \( \sin^2 x \) and \( \cos^4 x \) vary in such a way that the expression \( A = \sin^2 x + \cos^4 x \) remains between certain bounds for all real \( x \). Understanding these bounds requires exploring the range of functions, which is connected to how \( \sin \) and \( \cos \) cycle through their values as angles change.

Range of Functions

The range of a function is the set of all possible output values. For the sine and cosine functions, their ranges are particularly elegant.

• The values of \( \sin x \) and \( \cos x \) always lie between -1 and 1.
• Consequently, \( \sin^2 x \) and \( \cos^2 x \) values lie between 0 and 1.

In this problem, \( A = \sin^2 x + \cos^4 x \), a combination of these squared functions, involves analyzing how these values interact.

To find the range of \( A \) specifically, we note that since \( \cos^4 x \) or \( (1 - \sin^2 x)^2 \) must also be between 0 and 1 (as it is squaring a value from \( \cos^2 x \)), the behavior of \( A \) is closely tied to these constraints. \( A \)'s potential values come from the interplay of the ranges of both its sine square and cosine fourth-power components, yielding a possible range of \( \frac{3}{4} \leq A \leq 1 \). This is achieved by completing the square and determining the extremes of this expression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They serve as critical tools for simplifying expressions and solving trigonometric equations.

One of the foundational identities used in this problem is \( \sin^2 x + \cos^2 x = 1 \). This identity allows transforming and expressing various trigonometric properties and relationships. By reformulating the cosine square as \( \cos^2 x = 1 - \sin^2 x \), you can derive \( \cos^4 x = (1 - \sin^2 x)^2 \).

Additionally, completing the square method was utilized. Rewriting \( A = \sin^2 x + (1 - \sin^2 x)^2 \) leads effectively to \( A = (y - \frac{1}{2})^2 + \frac{3}{4} \), allowing easy evaluation of the range by leveraging trigonometric identities. Such transformations clarify the upper and lower bounds of function expressions and are vital for adequate range determination. These identities showcase the power of recognizing and applying fundamental trigonometric relationships to solve more complex expressions.