Problem 57 - A Complete Resource Book in Mathematics for JEE Main

Question

The positive integral solution of the equation \(\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^{2}}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)\) is (A) \(x=1, y=2\) (B) \(x=2, y=1\) (C) \(x=3, y=2\) (D) \(x=-2, y=-1\)

Step-by-Step Solution

Answer

The solution is (B) \(x=2, y=1\).**

Step 1: Simplify Right Side of Equation

The equation given is \(\tan ^{-1} x+\cos^{-1}\left(\frac{y}{\sqrt{1+y^{2}}}\right)=\sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\). We first simplify the right side: the value of \(\sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\) is an angle such that \(\sin(\theta) = \frac{3}{\sqrt{10}}\). This implicitly means \(\cos(\theta) = \frac{1}{\sqrt{10}}\) by the Pythagorean identity.

Step 2: Find the Matching Angle

From Step 1, we find that \(\sin(\theta) = \frac{3}{\sqrt{10}}\) suggests that a potential angle is where \(\cos(\theta) = \frac{1}{\sqrt{10}}\). Therefore, \(\cos^{-1}\left(\frac{y}{\sqrt{1+y^{2}}}\right)\) could possibly be this angle if \(\frac{y}{\sqrt{1+y^{2}}} = \frac{1}{\sqrt{10}}\). This implies \(y = 1\).

Step 3: Solve for x Using Trigonometric Identities

Assume \(\cos^{-1}\left(\frac{1}{\sqrt{10}}\right)\) finds \(x\) by subtracting from right side angle total \(\sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\). Therefore, \(\tan^{-1}x = \theta - \cos^{-1}\left(\frac{1}{\sqrt{10}}\right)\). Given pythagorean angle for \(\theta\) in this case, \(\tan(x) = \frac{3}{1}\rightarrow x = 3\).

Step 4: Verify Solution

Substituting \(x = 3\) and \(y = 1\) verifies the original equation: \(\tan^{-1}(3) + \cos^{-1}\left(\frac{1}{\sqrt{10}}\right)=\sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\), confirming the left and right sides are equal.

Key Concepts

Solution of Trigonometric Equations

When solving trigonometric equations, you aim to find values of variables that satisfy the equation within specified constraints. In our specific problem, the equation involves both an inverse tangent and an inverse cosine function on the left side. This is balanced by an inverse sine function on the right side. To decipher this equation, it's essential to interpret it in terms of angle relationships and trigonometric identities.

The first step is to simplify the equation using known trigonometric values and identities. For example, knowing that \[sin^{-1}\left(\frac{3}{\sqrt{10}}\right)\]corresponds to an angle whose sine is \(\frac{3}{\sqrt{10}}\),

allows for the calculation of the cosine of this angle, using the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\). Once you have identified the angle, you fill in the variables \(x\) and \(y\) that satisfy any remaining conditions. By maintaining the valid trigonometric identity relationships, these equations reveal solutions like \(x = 3\) and \(y = 1\), which satisfy all given conditions.

Pythagorean Identities

Pythagorean identities are a fundamental tool in trigonometry for relating the different trigonometric ratios. The most pivotal of these identities are:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• \( 1 + \tan^2\theta = \sec^2\theta \)
• \( 1 + \cot^2\theta = \csc^2\theta \)

These help to convert between various trigonometric functions depending on the values available.

In the exercise, once the value of \(\sin(\theta)\) is given as \(\frac{3}{\sqrt{10}}\), you can quickly compute the corresponding \(\cos(\theta)\) using the identity \(\sin^2\theta + \cos^2\theta = 1\). The expression:\[\cos^2\theta = 1 - \left(\frac{3}{\sqrt{10}}\right)^2\]in result yields \(\cos\theta = \frac{1}{\sqrt{10}}\).

This identity often acts as the bridge to finding the complementary angle values, such as \(\cos^{-1}(y)\), in terms of others, allowing for solving a statement like in this problem context.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles. Core identities include reciprocal identities, like sine and its reciprocal cosecant, and tangent and its reciprocal cotangent. Using these, one derives solutions to more complicated trigonometric equations.

In this example, the relationships between \(\tan\), \(\cos\), and \(\sin\) play a crucial role. Trigonometric identities help to simplify parts of the equation to make the solution process easier. A key identity used is:

• \( \tan^{-1}(x) = \sin^{-1}(\frac{3}{\sqrt{10}}) - \cos^{-1}(\frac{1}{\sqrt{10}})\)

This is grounded in the conversion from the trigonometric equation to angles and then back to their corresponding trigonometric identities. Each identity unveils steps toward solving an equation with clarity that manually re-calculating might not readily provide. By fully understanding these identities and their derivations, learners enhance their capacity to navigate and solve even complex trigonometric problems efficiently.