f 0 (α)/(β) (C) (tanα)/(tanβ)<(α)/(

Name: Problem 5, Chapter 14: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-09-20T02:54:41-08:00
Duration: 1 min 32 s
Channel: Mahipal Kumawat
Description: f 0 (α)/(β) (C) (tanα)/(tanβ) (α)/(β)

step 1 We are given that alpha is greater than 0 but less than beta and beta is less than pi by 2. That

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) $\frac{\tan \alpha}{\tan \beta}>\frac{\alpha}{\beta}$

   $\mathrm{f} 0<\alpha<\beta<\frac{\pi}{2}$ 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) $\frac{\tan \alpha}{\tan \beta}>\frac{\alpha}{\beta}$

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 14, Problem 5

Instant Answer

Solved by Expert Mahipal Kumawat

09/20/2021

Step 1

This means that $\alpha$ and $\beta$ belong to the interval $(0, \frac{\pi}{2})$. Show more…

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$\mathrm{f} 0<\alpha<\beta<\frac{\pi}{2}$ 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) $\frac{\tan \alpha}{\tan \beta}>\frac{\alpha}{\beta}$

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Key Concepts

Monotonicity of Trigonometric Functions

In the interval (0, ?/2), the tangent function is strictly increasing. This quality ensures that as the angle increases, tan(?) also increases, which is fundamental when comparing expressions involving tan(?) and tan(?) for 0 < ? < ? < ?/2.

Convexity of the Tangent Function

The tangent function is convex on the interval (0, ?/2). Convexity implies that the rate of increase of tan(?) grows as ? increases, which is instrumental in establishing inequalities involving ratios such as tan(?)/tan(?) compared to ?/?.

Behavior of the Function f(x)=tan(x)/x

The function f(x)=tan(x)/x is increasing on (0, ?/2) due to the properties of the tangent function. This increasing behavior allows one to compare the values at different points (? and ?) and derive inequalities by cross?multiplying, ultimately linking the ratios of tangents to the ratios of the angles.

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