Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\left|\cos \theta\left\\{\sin \theta+\sqrt{\sin ^{2} \theta+\sin ^{2} \alpha}\right\\}\right| \leq k\), then the value of \(k\) is (A) \(\sqrt{1+\cos ^{2} \alpha}\) (B) \(\sqrt{1+\sin ^{2} \alpha}\) (C) \(\sqrt{2+\sin ^{2} \alpha}\) (D) \(\sqrt{2+\cos ^{2} \alpha}\)

Step-by-Step Solution

Answer

The value of \( k \) is \( \sqrt{1+\sin^2\alpha} \) (Option B).

Step 1: Simplify and Analyze the Expression

Consider the expression \( \cos \theta\{\sin \theta + \sqrt{\sin^{2}\theta + \sin^{2}\alpha}\} \). First, substitute \( x = \sin \theta \), so the expression becomes \( \cos \theta (x + \sqrt{x^2 + \sin^2 \alpha}) \).

Step 2: Use Trigonometric Identities

Recall that \( \cos \theta = \sqrt{1-x^2} \). Substitute \( \cos \theta \) into the expression: \[ \sqrt{1-x^2} \left(x + \sqrt{x^2 + \sin^2 \alpha}\right) \]

Step 3: Understand the Maximum Value

For the expression \( \sqrt{1-x^2} \left(x + \sqrt{x^2 + \sin^2 \alpha}\right) \), the maximum value will occur when both \( x \) and \( \sqrt{x^2 + \sin^2 \alpha} \) contribute most significantly. Consider the extreme case of \( x = 1 \), which gives \[ \sqrt{0}(1 + \sqrt{1 + \sin^2 \alpha}) = 0 \] and isn't helpful. So analyze when \( x = 0 \).

Step 4: Evaluate with Simplified Conditions

If \( x = 0 \), the expression simplifies to \[ \cos \theta \sqrt{\sin^2 \alpha} = \sqrt{1} \cdot \sin \alpha = \sin \alpha \]. Then the expression to consider is when the non-zero term (i.e., the square root term) dominates.

Step 5: Evaluate Maximum Possible \( k \)

Re-evaluate for \( \sqrt{x^2 + \sin^2 \alpha} \) being maximized, so plug in extreme trigonometric values. If \( x = 0 \), then the expression becomes: \[ \sqrt{1-x^2} \cdot \sqrt{x^2 + \sin^2 \alpha} = 1 \times \sqrt{0 + \sin^2 \alpha} = \sqrt{\sin^2 \alpha} = 1 \].

Step 6: Check Trigonometric Value Consistency

Adjust the expression to test for full value range. The entire expression is less than or equal to: \[ \left|\cos \theta\right| \sqrt{1 + \sin^2 \alpha} = \sqrt{1 + \sin^2 \alpha} \] which matches maximum values when \( \sin \theta = 0 \) or \( \cos \theta = 1 \).

Step 7: Conclude with Solution Selection

By results from steps, confirm option where value of \( k \) equals derived formula, \( \sqrt{1+\sin^2\alpha} \) aligns with maximum conditions across chosen extremes.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial in simplifying complex expressions, especially when dealing with sine and cosine functions. These identities help us transform trigonometric expressions into alternative forms that are easier to analyze. For instance, we often use the Pythagorean identity:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

Substituting variables is a common approach. In our exercise, we substituted \( x \) for \( \sin \theta \) to simplify the expression. Knowing that \( \cos \theta = \sqrt{1-x^2} \) directly stems from the Pythagorean identity, illustrates the transformative power of these identities. By substituting more familiar terms from trigonometric identities, expressions become less intimidating.
Understanding these relationships helps eliminate complexities when solving trigonometric problems, allowing us to focus on calculating maximum values or other objectives of the exercise.

Maximum Value in Trigonometry

Finding maximum values in trigonometric expressions involves understanding how each trigonometric function behaves within its boundaries. The sine and cosine functions have values strictly between -1 and 1.
In the context of our solution, the expression was \[ \sqrt{1-x^2} \left(x + \sqrt{x^2 + \sin^2 \alpha}\right) \] The challenging part is determining when this expression hits its maximum. We initially tried extreme values, like setting \( x = 1 \) and \( x = 0 \). However, the expression yielded unhelpful results like zero when \( x = 1 \). Then concentrating on \( x = 0 \) resulted in interesting simplifying conditions, making it easier to tackle.

• With \( x = 0 \), we determined the maximum possible scenario using both sine, cosine, and Pythagorean identities to get \( \sqrt{1+\sin^2 \alpha} \).

Employing strategic value placements, alongside trigonometric identities, allows determination of a clear maximum without unnecessary complexity.

Problem-Solving Strategies in Mathematics

Mathematical problem-solving, especially in trigonometry, utilizes several key strategies. These strategies include making substitutions, identifying identities, and considering limits of function values. In our problem:

• We addressed the complexity by substituting \( \sin \theta \) with simpler symbolic terms.
• Implementing this substitution transformed a tricky problem into a more approachable form.
• Next was testing with known trigonometric extremes, such as setting \( x \) to boundary values of sine functions (0 or 1) to explore maximum contributions.
• Finally, re-evaluating the terms integrated in the expression led to identifying the maximum value based on already well-understood trigonometric limits.

By breaking down complex mathematics into logical and strategic steps, students can simplify any trigonometric challenge effectively. Developing this skill builds confidence and ensures a deep understanding, vital for tackling more advanced topics.