Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(0<\alpha<\beta<\frac{\pi}{2}\) then (A) \(\frac{\tan \beta}{\tan \alpha}<\frac{\alpha}{\beta}\) (B) \(\frac{\tan \beta}{\tan \alpha}>\frac{\alpha}{\beta}\) (C) \(\frac{\tan \alpha}{\tan \beta}<\frac{\alpha}{\beta}\) (D) \(\frac{\tan \alpha}{\tan \beta}>\frac{\alpha}{\beta}\)

Step-by-Step Solution

Answer

Option (B) \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) is correct.

Step 1: Understand the Given Condition

We are given that \(0 < \alpha < \beta < \frac{\pi}{2}\). This indicates that both \(\alpha\) and \(\beta\) are angles in the first quadrant where the tangent function is positive and increasing.

Step 2: Compare Tan Values

Since \(\tan(x)\) is an increasing function in the interval \( (0, \frac{\pi}{2})\), we know that \( \tan(\alpha) < \tan(\beta) \). This implies that \( \frac{\tan(\beta)}{\tan(\alpha)} > 1\).

Step 3: Analyze Given Options

Given the result from Step 2, option (B) \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) needs to be evaluated.

Step 4: Evaluate Option B

We know \( \alpha < \beta \) implies \( \frac{\alpha}{\beta} < 1\). Since \(\frac{\tan \beta}{\tan \alpha} > 1\) from Step 2, we conclude that \( \frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta} \). Thus, option (B) is correct.

Step 5: Verify Other Options

For completeness, verify the other options:- Option (A) \(\frac{\tan \beta}{\tan \alpha} < \frac{\alpha}{\beta}\) contradicts Step 4.- Option (C) and (D) involve \(\frac{\tan \alpha}{\tan \beta}\) which will be less than 1, conflicting with the given options suggesting inequality with \(\frac{\alpha}{\beta}\).

Key Concepts

Tangent Function

The tangent function is a fundamental component in trigonometry, especially when dealing with right triangles and circles. It expresses the ratio of the opposite side to the adjacent side in a right-angled triangle. Mathematically, it is written as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). The function is defined for all angles, but in the context of this problem, we focus on the interval \( (0, \frac{\pi}{2}) \), which is the first quadrant of the unit circle.

The tangent function is a continuous and increasing function within this quadrant. This means as the angle \( \theta \) increases from 0 to \( \frac{\pi}{2} \), the value of \( \tan(\theta) \) increases without bound. Understanding the behavior of \( \tan \) in the first quadrant allows us to predict how it behaves relative to other trigonometric functions such as sine and cosine.

• \( \tan(0) = 0 \)
• \( \tan(\frac{\pi}{4}) = 1 \)
• \( \tan(\theta) \to \infty \) as \( \theta \to \frac{\pi}{2} \)

This increasing nature is key in comparing different angles' tangent values, as seen in the exercise.

Inequality

Inequalities allow us to demonstrate the relative size between two expressions. They are crucial in mathematical reasoning and proofs. When we say \( a < b \), it implies that \( a \) is smaller than \( b \). Similarly, \( a > b \) means \( a \) is bigger than \( b \).

In trigonometry, inequalities help compare trigonometric values at different angles. For instance, knowing that the tangent function is increasing in the first quadrant, we can assert \( \tan(\alpha) < \tan(\beta) \) if \( \alpha < \beta \). This inequality holds because in the interval \( (0, \frac{\pi}{2}) \), \( \tan \) maps to positive numbers increasing with the angle.

• \( \frac{\tan(\beta)}{\tan(\alpha)} > 1 \) because \( \tan(\alpha) < \tan(\beta) \)
• \( \frac{\alpha}{\beta} < 1 \) under the condition \( \alpha < \beta \)

Understanding how these inequalities interact is essential to solve problems like the one given and find the correct relation between angles and their trigonometric ratios.

First Quadrant Angles

First quadrant angles are those angles measured from 0 to \( \frac{\pi}{2} \), or 0 to 90 degrees, and they have special properties in trigonometric contexts.

In this quadrant, all trigonometric functions — sine, cosine, and tangent — yield positive values. This positivity is why these angles are crucial in many applications, like physics and engineering, where non-negative results are often required.

• For any angle \( \theta \) in the first quadrant, \( 0 < \sin(\theta), \cos(\theta), \tan(\theta) \)
• Knowing the function's behavior, \( \tan \) here is crucial; it’s increasing, meaning that larger angles produce larger tangent values.

First quadrant angles help simplify many trigonometric problems. Recognizing that the tangent of any two angles in this range fulfills \( \tan(\alpha) < \tan(\beta) \) if \( \alpha < \beta \) is key to understanding the core comparison in the exercise.