Problem 28 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\sin ^{-1}\left(\frac{x}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}\), then the values of \(\mathrm{x}\) is (a) 4 (b) \(5 \) (c) 1 (d) 3

Step-by-Step Solution

Answer

The value of \(x\) is 4, choice (a).

Step 1: Understand the Exercise

The problem involves inverse trigonometric functions and a sum of angles equation. We need to find the value of \( x \) that satisfies the equation: \( \sin^{-1}\left(\frac{x}{5}\right) + \operatorname{cosec}^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2} \).

Step 2: Rewrite Using Complementary Angles

Since \( \sin^{-1}(a) + \csc^{-1}(b) = \frac{\pi}{2} \) typically leads to complementary functions, we can use the identity: \( \csc^{-1}\left(\frac{a}{b}\right) = \sin^{-1}(\frac{b}{a}) \). Apply it here: \( \sin^{-1}\left(\frac{5}{4}\right) \) is equivalent to \( \csc^{-1}\left(\frac{5}{4}\right).\)

Step 3: Equalize Sin Inverses

With the conversion, rewrite the equation as \( \sin^{-1}\left(\frac{x}{5}\right) = \sin^{-1}\left(\frac{4}{5}\right) \).

Step 4: Solve for x

If \( \sin^{-1}\left(\frac{x}{5}\right) = \sin^{-1}\left(\frac{4}{5}\right) \), equate their arguments: \( \frac{x}{5} = \frac{4}{5} \). Multiplying both sides by 5 gives \( x = 4 \).

Step 5: Verify with Choice Options

Compare the obtained solution with the provided choices: (a) 4, (b) 5, (c) 1, (d) 3. The value \( x = 4 \) matches choice (a).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical relationships that involve trigonometric functions, such as sine, cosine, tangent, and their inverses. These identities help us solve complex equations by transforming or simplifying expressions. One fundamental identity is the complementary angle identity, which states that the sum of certain pairs of angles equal to \( \frac{\pi}{2} \) or 90 degrees.

For example, one such identity is:

• \( \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \)
• \( \csc^{-1}(a) + \sec^{-1}(a) = \frac{\pi}{2} \)

These identities are particularly useful when dealing with inverse trigonometric functions because they provide a way to convert one function into another, aiding in solving equations. For instance, in the given exercise, the identity used is \( \csc^{-1}\left(\frac{a}{b}\right) = \sin^{-1}\left(\frac{b}{a}\right) \), which is derived from the concept of complementary angles.

Complementary Angles

Complementary angles are pairs of angles whose measures sum up to 90 degrees or \( \frac{\pi}{2} \) in radians. This concept is crucial when working with inverse trigonometric functions, as it allows the transformation between certain trigonometric expressions.

Understanding complementary angles provides a straightforward way to simplify equations. This is because if one angle is known, the other is implicitly defined by this relationship.
For example:

• If you have an angle \( \theta \), its complement is \( \frac{\pi}{2} - \theta \).
• In the case of inverse functions, such as \( \sin^{-1}(a) \) and \( \csc^{-1}(a) \), these two can be related by their complementary nature.

In the problem at hand, recognizing that the sum \( \sin^{-1}\left(\frac{x}{5}\right) + \operatorname{cosec}^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2} \) indicates that these are complementary angles, is key to solving the exercise.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions and their relationships. Solving these equations generally requires applying identities or transformations to simplify and find the values of unknown variables.

In the context of the given exercise, the goal is to find the value of \( x \) that satisfies the angle sum equation. By identifying the relationship between the given functions as complementary, we can rewrite the equation effectively.

• First, use the identity \( \csc^{-1}(a) = \sin^{-1}\left(\frac{1}{a}\right) \) to relate the trigonometric terms.
• Second, once converted, the equation \( \sin^{-1}\left(\frac{x}{5}\right) = \sin^{-1}\left(\frac{4}{5}\right) \) follows, which implies that \( \frac{x}{5} = \frac{4}{5} \).
• Solving for \( x \) becomes straightforward—multiply both sides by 5 to isolate \( x \), resulting in \( x = 4 \).

By systematically applying the identities and understanding the properties of inverse functions, solving trigonometric equations becomes manageable and insightful.