Problem 79 - A Complete Resource Book in Mathematics for JEE Main

Question

Assertion: \(\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}=\frac{\pi}{2}\) Reason: \(\sin ^{-1} x+\sin ^{-1} y\) \(=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\)

Step-by-Step Solution

Answer

The assertion is true because \( \sin^{-1} \frac{4}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{16}{65} = \frac{\pi}{2} \).

Step 1: Understand the Expression

Given the assertion, we have to prove that \( \sin^{-1} \frac{4}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1}\frac{16}{65} = \frac{\pi}{2} \). The reason provides a formula for the sum of two arcsine terms.

Step 2: Apply the Formula to First Two Terms

Using the formula \( \sin^{-1}x + \sin^{-1}y = \sin^{-1}(x \sqrt{1 - y^2} + y \sqrt{1 - x^2}) \), we set \( x = \frac{4}{5} \) and \( y = \frac{5}{13} \). Calculating:\[1 - \left(\frac{5}{13}\right)^2 = \frac{144}{169}, \quad 1 - \left(\frac{4}{5}\right)^2 = \frac{9}{25}\]Substituting in, we have:\[\sin^{-1}\left( \frac{4}{5} \cdot \frac{12}{13} + \frac{5}{13} \cdot \frac{3}{5} \right) = \sin^{-1}\left(\frac{48}{65} + \frac{15}{65}\right) = \sin^{-1}\left(\frac{63}{65}\right)\]

Step 3: Combine Resulting Term with Third Term

Now, use the arcsine sum formula to add \(\sin^{-1}\left(\frac{63}{65}\right)\) and \(\sin^{-1}\left(\frac{16}{65}\right)\). Let \(x = \frac{63}{65}\) and \(y = \frac{16}{65}\):\[1 - \left(\frac{16}{65}\right)^2 = \frac{3969}{4225}, \quad 1 - \left(\frac{63}{65}\right)^2 = \frac{256}{4225}\]Then we combine:\[\sin^{-1}\left( \frac{63}{65} \cdot \frac{63}{65} + \frac{16}{65} \cdot \frac{16}{65} \right) = \sin^{-1}\left(\frac{3969}{4225} + \frac{256}{4225}\right) = \sin^{-1}(1)\]Since \(\sin^{-1}(1) = \frac{\pi}{2}\), the assertion is true.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions provide a way to determine the angles whose trigonometric ratios are known. These functions are the inverse of the regular trigonometric functions, such as sine, cosine, and tangent. Each of these functions has its own domain and range, which are important to remember when working with inverse functions.

• For example, \(\sin^{-1}(x)\) or arcsin is the angle \(\theta\) such that \(\sin(\theta) = x\).
• The typical range for \(\sin^{-1}(x)\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Understanding these ranges and domains is crucial, as it helps solve equations like the one in the exercise. The operation of adding angles using their inverse trigonometric functions, like \(\sin^{-1} x + \sin^{-1} y = \sin^{-1}(x \sqrt{1-y^2} + y \sqrt{1-x^2})\), is based on these principles.

Sum of Arcsine

The sum of arcsine involves understanding how to combine two inverse sine values into a single expression. The formula utilized for this in the exercise allows us to add two arcsine functions efficiently:
\[ \sin^{-1} x + \sin^{-1} y = \sin^{-1}(x \sqrt{1-y^2} + y \sqrt{1-x^2}) \]\ This formula is derived from the basic properties of trigonometric and inverse trigonometric functions, where vectors in a unit circle are often involved. When solving problems like the one given, you start by substituting the specific values you have into the formula.
This method is particularly useful in scenarios like competitive exams, such as JEE Main Mathematics, where time efficiency and accuracy are key. By breaking the problem into smaller parts using such identities, we can verify results or derive new conclusions about trigonometric equations.

JEE Main Mathematics

The JEE Main Mathematics examination often includes questions on trigonometric identities and inverse trigonometric functions. A strong grasp of these topics is essential because:

• They test students' understanding of mathematical concepts.
• They evaluate problem-solving skills under time constraints.

Understanding basic formulas and identities thoroughly helps tackle complex questions by simplifying them into manageable parts. In inverse trigonometric functions, recognizing when and why to use specific identities is crucial.
For instance, in the exercise given, knowing the arcsine addition formula allows students to solve the problem more rapidly, thereby improving efficiency. This strategic application of knowledge is what makes a notable difference in competitive math examinations like JEE Main Mathematics, where analytical skills and speed are equally important.