Problem 31 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\frac{\sin \alpha}{\sin \beta}=\frac{\sqrt{3}}{2}\) and \(\frac{\cos \alpha}{\cos \beta}=\frac{\sqrt{5}}{2}, 0<\alpha<\beta<\frac{\pi}{2}\), then (A) \(\tan \alpha=1\) (B) \(\tan \alpha=\frac{\sqrt{3}}{\sqrt{5}}\) (C) \(\tan \beta=\frac{\sqrt{3}}{\sqrt{5}}\) (D) \(\tan \beta=1\)

Step-by-Step Solution

Answer

Option (B): \( \tan \alpha = \frac{\sqrt{3}}{\sqrt{5}} \).

Step 1: Understand the given equations

We have two given ratios: \( \frac{\sin \alpha}{\sin \beta} = \frac{\sqrt{3}}{2} \) and \( \frac{\cos \alpha}{\cos \beta} = \frac{\sqrt{5}}{2} \). Our task is to find either \( \tan \alpha \) or \( \tan \beta \) using these equations.

Step 2: Express sine and cosine in terms of each other

Using the identity \( \tan x = \frac{\sin x}{\cos x} \), express \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} \).

Step 3: Express sin and cos in terms of \( \alpha \)

From \( \frac{\sin \alpha}{\sin \beta} = \frac{\sqrt{3}}{2} \), we have \( \sin \alpha = \frac{\sqrt{3}}{2} \sin \beta \). Similarly, from \( \frac{\cos \alpha}{\cos \beta} = \frac{\sqrt{5}}{2} \), we have \( \cos \alpha = \frac{\sqrt{5}}{2} \cos \beta \).

Step 4: Substitute into \( \tan \alpha \)

Express \( \tan \alpha \) as \( \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{\sqrt{3}}{2} \sin \beta}{\frac{\sqrt{5}}{2} \cos \beta} \). Simplifying gives \( \tan \alpha = \frac{\sqrt{3}}{\sqrt{5}} \).

Step 5: Determine \( \tan \beta \)

Similarly, for \( \tan \beta \), express it as \( \frac{\sin \beta}{\cos \beta} \). Since neither \( \sin \beta \) nor \( \cos \beta \) were explicitly altered, \( \tan \beta = \tan \beta \) remains as is.

Step 6: Match with given options

Checking the calculated values for \( \tan \alpha \) and \( \tan \beta \) against the options: \( \tan \alpha = \frac{\sqrt{3}}{\sqrt{5}} \) matches with option (B).

Key Concepts

Sine and Cosine Ratios

Sine and cosine are fundamental trigonometric functions that relate the angles and sides of a right triangle. These functions are crucial in understanding circular and oscillatory movements. In every right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse.
For example, if we denote an angle by \( \alpha \), then

• Sine Ratio: \( \sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• Cosine Ratio: \( \cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

In the given exercise, these ratios help us compare the angles \( \alpha \) and \( \beta \) by solving equations such as \( \frac{\sin \alpha}{\sin \beta} = \frac{\sqrt{3}}{2} \) and \( \frac{\cos \alpha}{\cos \beta} = \frac{\sqrt{5}}{2} \). By manipulating these expressions, we can deduce further properties like the tangent of an angle.

Tangent Function

The tangent function is another essential trigonometric ratio. It is defined as the ratio of the sine of an angle to its cosine. If we consider an angle \( x \), the tangent can be represented as:

• \( \tan x = \frac{\sin x}{\cos x} \)

This function is incredibly useful when it comes to finding angles or solving triangles, especially when the lengths of opposite and adjacent sides are involved.
In our example, we use the concept of the tangent function to find \( \tan \alpha \). Using the expressions derived for sine and cosine, \( \tan \alpha = \frac{\frac{\sqrt{3}}{2} \sin \beta}{\frac{\sqrt{5}}{2} \cos \beta} \), simplifying this gives \( \tan \alpha = \frac{\sqrt{3}}{\sqrt{5}} \). Hence, the tangent function provides a way to relate all the information we have from sine and cosine ratios.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variable involved. These identities form the backbone of trigonometry, providing tools for simplifying and solving a wide variety of problems.
A particularly useful identity is the reciprocal identity that relates the tangent function to sine and cosine:

• \( \tan x = \frac{\sin x}{\cos x} \)

These identities allow transformations of angles and variables for simplifying calculations. For example, with given \( \frac{\sin \alpha}{\sin \beta} \) and \( \frac{\cos \alpha}{\cos \beta} \) ratios, trigonometric identities help us transition these to a single function like the tangent, supporting us in finding the final answer in the problem.
Understanding these identities and how they connect different trigonometric functions is crucial for solving complex exercises efficiently and accurately.