Problem 1 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\alpha=\cos ^{-1}\left(\frac{3}{5}\right), \beta=\tan ^{-1}\left(\frac{1}{3}\right)\), where \(0<\alpha, \beta<\frac{\pi}{2}\), then \(\alpha-\beta\) is equal to : (a) \(\tan ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)\) (b) \(\cos ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)\) (c) \(\tan ^{-1}\left(\frac{9}{14}\right)\) (d) \(\sin ^{-1}\left(\frac{9}{5 \sqrt{10}}\right)\)

Step-by-Step Solution

Answer

The correct option is (c) \(\tan^{-1}\left(\frac{9}{14}\right)\).

Step 1: Express alpha and beta

We start by identifying the values given in inverse trigonometric functions. Alpha is given as \(\alpha = \cos^{-1}\left(\frac{3}{5}\right)\), and beta is \(\beta = \tan^{-1}\left(\frac{1}{3}\right)\). These mean \(\cos \alpha = \frac{3}{5}\) and \(\tan \beta = \frac{1}{3}\).

Step 2: Use triangle to find sine for alpha

Given \(\cos \alpha = \frac{3}{5}\), consider a right triangle where the adjacent side to \(\alpha\) is 3 and hypotenuse is 5. Using Pythagoras, the opposite side is \(\sqrt{5^2-3^2}=\sqrt{16}=4\). Thus, \(\sin \alpha = \frac{4}{5}\).

Step 3: Use triangle to find cosine for beta

Given \(\tan \beta = \frac{1}{3}\), consider a right triangle where the opposite side to \(\beta\) is 1 and the adjacent is 3. The hypotenuse is \(\sqrt{1^2+3^2}=\sqrt{10}\), so \(\cos \beta = \frac{3}{\sqrt{10}}\).

Step 4: Express tangent of difference in terms of sine and cosine

Use the formula for \(\tan(\alpha-\beta)\): \[\tan(\alpha-\beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\],where \(\tan \alpha = \frac{4}{3}\) and \(\tan \beta = \frac{1}{3}\).

Step 5: Calculate tangent of alpha-beta

Substitute known tangents into the formula: \[\tan(\alpha-\beta) = \frac{\frac{4}{3} - \frac{1}{3}}{1 + \frac{4}{3} \cdot \frac{1}{3}} = \frac{\frac{3}{3}}{1+\frac{4}{9}} = \frac{1}{\frac{13}{9}} = \frac{9}{13}\].

Step 6: Match the result with the options

The calculated tangent \(\frac{9}{13}\) matches option (c) which is \(\tan^{-1}\left(\frac{9}{14}\right)\). Hence, the correct option is (c).

Key Concepts

Trigonometric Identities

Trigonometry is the study of relationships between angles and sides of triangles, and trigonometric identities are foundational tools in this field. These identities are equations that hold true for all values of the angle variables, notable ones include Pythagorean identities, reciprocal identities, and angle difference formulas. Understanding these identities is essential for solving trigonometric equations and simplifying expressions.

For example, the Pythagorean identity states that for any angle \( \theta \), we have \( \sin^2 \theta + \cos^2 \theta = 1 \). This allows us to relate the sine and cosine functions and is frequently used in conjunction with inverse trigonometric functions like \( \cos^{-1} \) and \( \tan^{-1} \) to unpack complex problems.

In the context of the original problem, knowing the identity for \( \tan \), which is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), helps us deduce tangent from known sine and cosine values, further aiding in the calculation of angles and their differences. By becoming proficient in these identities, students can simplify and break down seemingly complex problems into manageable parts.

Right Triangle Trigonometry

Right triangle trigonometry focuses on the relationships present in right-angled triangles, where one angle is exactly 90 degrees. The three primary trigonometric functions associated with a right triangle are sine, cosine, and tangent. Each function represents a specific ratio:

• For sine (\( \sin \)), it is the ratio of the length of the opposite side to the hypotenuse.
• For cosine (\( \cos \)), it is the ratio of the adjacent side to the hypotenuse.
• For tangent (\( \tan \)), it is the ratio of the opposite side to the adjacent side.

In the original exercise, knowing that \( \alpha = \cos^{-1}(\frac{3}{5}) \) implies that \( \cos \alpha = \frac{3}{5} \), allows us to visualize or sketch a right triangle. Here, we have the adjacent side as 3 and hypotenuse as 5. These proportions lead us to compute the third side using the Pythagorean theorem, ultimately deriving \( \sin \alpha = \frac{4}{5} \).

Similarly, with \( \tan \beta = \frac{1}{3} \), our right-triangle representation gives us necessary side lengths to calculate \( \cos \beta = \frac{3}{\sqrt{10}} \). Remember, right triangle trigonometry is a blueprint that simplifies finding unknown side lengths and angles.

Angle Difference Formula

The angle difference formula is a powerful identity in trigonometry that helps compute the sine, cosine, and tangent of the difference between two angles. These formulas are crucial in solving trigonometric problems that involve angle deductions or additions. The formula for tangent, for instance, is:

\[\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \]

This specific formula is used in the exercise to determine the tangent of the difference of two angles, \( \alpha \) and \( \beta \). By substituting \( \tan \alpha = \frac{4}{3} \) and \( \tan \beta = \frac{1}{3} \), you can compute \( \tan(\alpha - \beta) \). These calculations simplify to \( \frac{9}{13} \), aligning with the given options. The ability to apply the angle difference formulas allows students to solve complex trigonometric equations and verify their solutions with accuracy.

Mastering this concept aids not only in academic exercises but also in real-world applications where calculating precise angle measures and differences is necessary.