Problem 28 - A Complete Resource Book in Mathematics for JEE Main

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) \(\underline{B}\) (B) \(A\) (C) \(C\) (D) none of these

Step-by-Step Solution

Answer

A is greater than B, so the answer is (B) A.

Step 1: Simplify Angle A

First, simplify the expression for angle \( A = 2 \tan ^{-1}(2\sqrt{2} - 1) \). Let \( x = 2\sqrt{2} - 1 \). Then, \( \tan^{-1}x \) represents an angle whose tangent is \( x \). So, we need to evaluate this inverse tangent. Calculating gives \( x \approx 1.83 \), and thus \( \tan^{-1}(1.83) \approx 1.07 \) radians. Therefore, \( A = 2 \times 1.07 \approx 2.14 \) radians.

Step 2: Simplify Angle B

For angle \( B = 3\sin^{-1}\frac{1}{3} + \sin^{-1}\frac{3}{5} \), find each part individually. \(3\sin^{-1}\frac{1}{3}\) involves finding \( \sin^{-1} \frac{1}{3} \), which is an angle whose sine is \( \frac{1}{3} \). This gives \( \sin^{-1}\frac{1}{3} \approx 0.3398 \) radians. Therefore, \( 3\sin^{-1}\frac{1}{3} \approx 1.0194 \) radians. For the second part, \( \sin^{-1}\frac{3}{5} \approx 0.6435 \) radians. Thus, the total \( B = 1.0194 + 0.6435 \approx 1.6629 \) radians.

Step 3: Compare the Angles

Now compare the values of \( A \) and \( B \). From the previous steps, \( A \approx 2.14 \) radians and \( B \approx 1.6629 \) radians. Since \( 2.14 > 1.6629 \), angle \( A \) is greater than angle \( B \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions offer a way to determine an angle when you know the trigonometric value. They include

• \( \sin^{-1} \) (also known as arcsine)
• \( \cos^{-1} \) (arccosine)
• \( \tan^{-1} \) (arctangent)

Each function gives you an angle in radians or degrees.
For example, in our exercise, we encounter \( \tan^{-1}(x) \) and \( \sin^{-1}(x) \). This means we want to find the angle whose tangent or sine equals \( x \).

If we know that the tangent of an angle is 1.83, then using \( \tan^{-1} \), we determine the angle to be approximately 1.07 radians. This inverse process is crucial in many geometric computations.

Angle Comparison

Comparison of angles involves evaluating their measures, often given in radians or degrees. Angles are compared to determine which is greater or if they are equal. For the given exercise, we simplify each expression to find out the size of each angle in radians.

After simplifying, we derive that:

• Angle \( A \) is approximately 2.14 radians.
• Angle \( B \) is approximately 1.6629 radians.

This straightforward comparison shows that 2.14 is indeed greater than 1.6629, demonstrating that angle \( A \) is larger than angle \( B \).
This principle of comparing angle measures is often used in trigonometry to solve problems and can be visualized on a unit circle, where larger angles sweep out more of the circle.

Radian Measure

A radian is a unit of measure for angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius. It is a natural way to measure angles as it relates directly to the geometry of circles.

In the context of our exercise, both angles \( A \) and \( B \) are measured in radians. The radian measure is crucial because it allows us to use trigonometric functions effectively.
For example, when we compute \( \tan^{-1}(1.83) \), the result is automatically in radians. Radians are generally preferred in calculus and advanced mathematics because they simplify many formulas and calculations.

• One full circle (360 degrees) corresponds to \( 2\pi \) radians.
• This conversion is heavily utilized in various scientific fields.

Remember that understanding radian measure makes it easier to apply trigonometric functions to solve problems efficiently.