Problem 48 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The greater of the two angles \(A=2 \tan ^{-1}(2 \sqrt{2}-1)\) and \(B=3 \sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{3}{5}\) is (A) \(B\) (B) \(A\) (C) \(C\) (D) none of these

Step-by-Step Solution

Answer

The greater angle is A. Option (B) is correct.

Step 1: Analyze Angle A

The angle is defined as \( A = 2 \tan^{-1}(2\sqrt{2} - 1) \). We need to calculate \( \tan^{-1}(2\sqrt{2} - 1) \) and then double it. Using a calculator or table, we find that \( 2\sqrt{2} - 1 \approx 1.828 \), which corresponds to \( \tan^{-1}(1.828) \approx 61.93^\circ \). Therefore, \( A = 2 \times 61.93^\circ = 123.86^\circ \).

Step 2: Analyze Angle B

The angle is defined as \( B = 3 \sin^{-1} \frac{1}{3} + \sin^{-1} \frac{3}{5} \). First, calculate \( \sin^{-1} \frac{1}{3} \approx 19.47^\circ \) and \( \sin^{-1} \frac{3}{5} \approx 36.87^\circ \). Then calculate the whole expression, \( B = 3 \times 19.47^\circ + 36.87^\circ = 58.41^\circ + 36.87^\circ = 95.28^\circ \).

Step 3: Compare the Angles

Now, compare the results from Step 1 and Step 2. We have \( A = 123.86^\circ \) and \( B = 95.28^\circ \). It's clear that \( A > B \).

Step 4: Choose the Correct Answer

Since \( A \) is greater than \( B \), the correct answer is \( A \). Therefore, the answer choice is (B).

Key Concepts

Inverse Trigonometric Functions

The concept of inverse trigonometric functions is central to many trigonometric calculations. Inverse trigonometric functions allow us to determine angles from the lengths of the sides of a right triangle. For example:

• \( \sin^{-1} \) (or arcsin) is used to find an angle whose sine is a known value.
• \( \tan^{-1} \) (or arctan) finds an angle whose tangent is a known value.

For the angle \( A \) in the problem, the value \( \tan^{-1}(2\sqrt{2} - 1) \) gives us the angle whose tangent is \( 2\sqrt{2} - 1 \). To resolve this, you would typically use a scientific calculator or a trigonometric table, which converts this tangent value into an angle. In this exercise, \( \tan^{-1}(1.828) \approx 61.93^\circ \), which is nearly \( 62^\circ \). Since \( A = 2 \times 61.93^\circ \), doubling this angle gets us around \( 123.86^\circ \). Having a firm grasp on these functions helps in correctly solving trigonometric equations and comparing angles as done in this solution.

Angle Comparison

Comparing angles is essential when determining which angle is larger or smaller in a trigonometric context. In the given exercise:- Angle \( A = 123.86^\circ \)- Angle \( B = 95.28^\circ \)To decide which angle is greater, we look at their measures directly. A comparison confirms that \( A \), at \( 123.86^\circ \), is greater than \( B \), at \( 95.28^\circ \). When comparing angles, one helpful tip is to consider the magnitude of each angle. An angle larger than \( 90^\circ \) is recognized as obtuse, and an angle less than \( 90^\circ \) is acute. This method of analysis helps understand why \( A \) is significantly larger than \( B \). Thus, practicing such calculations helps in recognizing this difference quickly, helping in similar exercises.

Mathematical Calculations

Mathematical calculations involving inverse trigonometric functions demand precision and sometimes the aid of technology. Here's a recap of the calculations for clarity:- For \( A \), we calculate \( 2 \times \tan^{-1}(2\sqrt{2} - 1) \), giving us an angle of \( 123.86^\circ \).- For \( B \), we calculate \( 3 \times \sin^{-1} \frac{1}{3} + \sin^{-1} \frac{3}{5} \).This is found by evaluating each part step-by-step:1. \( \sin^{-1} \frac{1}{3} \approx 19.47^\circ \)2. \( \sin^{-1} \frac{3}{5} \approx 36.87^\circ \)3. Multiply and add to get \( 58.41^\circ + 36.87^\circ = 95.28^\circ \).The process of accurately performing these calculations with a calculator or trig table is critical. Each step builds upon the previous, ensuring that rounding errors are minimized and the most accurate value is used in subsequent calculations. Understanding these steps ensures proficiency in handling similar mathematical problems.