Problem 48 - 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) \(B\) (B) \(A\) (C) \(\bar{C}\) (D) none of these

Step-by-Step Solution

Answer

The greater angle is B.

Step 1: Simplify Angle A

To simplify angle \( A = 2 \tan^{-1}(2\sqrt{2} - 1) \), we must use a trigonometric identity. The identity \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \) is useful here. Set \( x = 2\sqrt{2} - 1 \) such that \( \tan(\theta) = x \) and calculate \( \tan(2 \times \tan^{-1}(x)) \). Solving gives us \( \tan A = 7 - 4\sqrt{2} \), and using trigonometric tables or calculators, we find that angle \( A \) is approximately 0.9828 radians.

Step 2: Simplify Angle B

For angle \( B = 3\sin^{-1} \left( \frac{1}{3} \right) + \sin^{-1} \left( \frac{3}{5} \right) \), it is helpful to know the values of \( \sin^{-1}(\frac{1}{3}) \) and \( \sin^{-1}(\frac{3}{5}) \). These values are approximately 0.3398 radians and 0.6435 radians, respectively. Then, \( 3 \sin^{-1} \left(\frac{1}{3}\right) \approx 1.0194 \). Adding these values gives \( B \approx 1.6629 \) radians.

Step 3: Compare Angles A and B

We need to compare the two angles. From Step 1, \( A \approx 0.9828 \) radians. From Step 2, \( B \approx 1.6629 \) radians. Clearly, \( B \) is greater than \( A \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are very important in solving various mathematical problems, especially those involving angles. They help us find angles when we know the trigonometric ratios.
For example, if we have a value for which a sine, cosine, or tangent was found, using inverse functions, we calculate the associated angle.
These functions are commonly represented as

• \( an^{-1}(x) \) for the inverse tangent
• \( ext{arc sine, written as } \ \ \sin^{-1}(x) \) for the inverse sine
• \( ext{arc cosine, or } \ \ \cos^{-1}(x) \) for inverse cosine

Let's take the example of the angle \( A = 2 \tan^{-1}(2\sqrt{2} - 1) \). First, we simplify this using a known identity for tangent of a double angle: \[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \], which helps us solve the angle from the given expression.

Angle Comparison

Comparing angles is a common task in trigonometry. Once the angles are converted to a common measure, like radians, they can be directly compared.
In trigonometric exercises, one of the main objectives is to determine which angle is larger.
In our example, we had two angles to compare: \( A = 0.9828 \) radians and \( B = 1.6629 \) radians. By simple numeric comparison, it is clear that angle \( B \) is greater than angle \( A \).
In general, always follow these simple steps when comparing angles:

• Convert to the same unit, radians or degrees.
• Simplify each angle if necessary using trigonometric identities or inverse functions.
• Check the resulting values against each other.

Radian Measure

Radians are an alternative to degrees for measuring angles and are often the preferred unit in calculus and higher mathematics.
A radian is the angle formed when the arc length of a circle is equal to the radius of the circle.
One of the main advantages is that they relate directly to the arc length, making it easier to work with circular functions and calculus.
Key things to remember:

• Radians and degrees can be converted back and forth using the conversion factors
  • \( 180^\circ = \pi \: \ \ \text{radians} \)
  • \( \text{and} \, 1 \, \ \ \text{radian} = \frac{180}{\pi}^\circ \)
• For precise work in mathematics, radian measure is usually more useful than degrees.

In our example, we evaluated angles in radians for accurate comparison.

Simplification of Expressions

Simplifying expressions in trigonometry can often involve using known identities or inverse trigonometric values to break down complex equations.
This step is crucial for both calculation and understanding of the underlying relationships.
In our original problem, angle \( A \) was simplified using the identity \( \tan(2\theta) \) to neatly express the double angle in terms of one radian measure.
Steps for simplification include:

• Identify which formulas or identities apply (e.g., double angle, sum, and difference identities).
• Substitute known values or expressions where possible.
• Calculate the simplified form step by step, checking each line carefully.

By following structured approaches like this, one can solve even complex trigonometric problems more easily.