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

Question

\(\mathrm{f} 0<\alpha<\beta<\frac{\pi}{2}\) 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): \( \frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta} \).

Step 1: Understand the Inequalities

The inequality involves comparing two fractions \( \frac{\tan \beta}{\tan \alpha} \) and \( \frac{\alpha}{\beta} \). We need to determine the relationship between \( \tan \alpha \), \( \tan \beta \), \( \alpha \), and \( \beta \), knowing that \( 0 < \alpha < \beta < \frac{\pi}{2} \).

Step 2: Analyze the Behavior of the Tan Function

Since the function \( \tan \theta \) is increasing in the interval \( (0, \frac{\pi}{2}) \), we conclude that \( \tan \alpha < \tan \beta \) whenever \( \alpha < \beta \). This fact is fundamental to the inequality comparisons.

Step 3: Compare the Two Fractions

We need to compare \( \frac{\tan \beta}{\tan \alpha} \) and \( \frac{\alpha}{\beta} \). Since \( \tan \beta > \tan \alpha \) (because \( \beta > \alpha \)), the fraction \( \frac{\tan \beta}{\tan \alpha} \) is greater than 1, whereas \( \frac{\alpha}{\beta} < 1 \) since \( \alpha < \beta \). Thus, \( \frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta} \).

Step 4: Choose the Correct Option

With the conclusion from Step 3, we determine that the correct inequality is \( \frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta} \), which corresponds to option (B).

Key Concepts

Tan Function Behavior

The tangent function, denoted as \(\tan \theta\), plays an integral role in trigonometry. One crucial aspect to understand is that it is an increasing function in the interval \((0, \frac{\pi}{2})\). This means that as the angle \(\theta\) increases within this range, the value of \(\tan \theta\) also increases. This behavior is especially important when looking at inequalities, such as the relationship between \(\tan \alpha\) and \(\tan \beta\), given that \(\alpha<\beta\).

For any angles \(0 < \alpha < \beta < \frac{\pi}{2}\), we know that:

• \(\tan \alpha < \tan \beta\)
• This implies the function's increasing nature as the angle grows nearer to \(\frac{\pi}{2}\)

Understanding this behavior helps in solving inequalities involving tangent, as it gives us a predictable pattern based on angle order.

Inequality Comparisons

When tackling trigonometric inequalities, especially like those involving tangent, the key lies in comparing the relationships between ratios or fractions. In the given problem, we compare \(\frac{\tan \beta}{\tan \alpha}\) with \(\frac{\alpha}{\beta}\).

Here are some steps to consider in such comparisons:

• First, recognize that since \(\tan \beta > \tan \alpha\) for \(\beta > \alpha\), \(\frac{\tan \beta}{\tan \alpha} > 1\).
• Secondly, because \(\frac{\alpha}{\beta} < 1\) (as \(\alpha < \beta\)), you're comparing something greater than 1 with something less than 1.
• This establishes that \(\frac{\tan \beta}{\tan \alpha}\) is clearly greater than \(\frac{\alpha}{\beta}\), affirming an important inequality difference due to the non-linear growth of the tan function compared to linear angles.

Seeing inequalities through fraction comparisons provides a clear, logical path to understand which side is greater or lesser.

Angle Relationships in Trigonometry

Understanding angle relationships is fundamental in trigonometry especially when you're dealing with trigonometric functions like tangent. For angles \(\alpha\) and \(\beta\) where \(0 < \alpha < \beta < \frac{\pi}{2}\), several relationships become apparent.

Here's what we know about these angles:

• All angles within \((0, \frac{\pi}{2})\) are positive, making trigonometric functions like tangent, positive too.
• Since \(\alpha < \beta\), geometrically, this implies that the line segment associated with \(\beta\) is longer than that associated with \(\alpha\).
• This length difference influences the tangent values, making \(\tan \beta > \tan \alpha\), directly tied to how these angles and their trigonometric functions relate to one another.

Recognizing such relationships simplifies the process of solving and understanding more complex trigonometric expressions and inequalities.