INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

May 6, 2014

An inequality related to (sin x)/x function | ISI BMath 2007

This is a problem number 7 from ISI B.Math 2007 based on an inequality related to (sin x)/x function. Try out this problem.

Problem: An inequality related to (sin x)/x function

Let $ \mathbf{0\leq \theta\leq \frac{\pi}{2}}$ . Prove that $\mathbf{\sin \theta \geq \frac{2\theta}{\pi}}$.


We consider the function $ \mathbf{ f(x) = \frac{\sin x }{x} } $. The first derivative of this function is $ \mathbf{ f'(x) = \frac{x \cos x - \sin x} {x^2} }$ In the interval $ \mathbf{[0, \frac{\pi}{2}]}$ the numerator is always negative as x is less than tan x.

Hence f(x) is a monotonically decreasing function in the given interval. Hence f(x) attains least value at $\mathbf{x = \frac{\pi}{2} }$ which equals $ \mathbf{ \frac{\sin\frac{\pi}{2}}{\frac{\pi}{2}} = \frac {2}{\pi}}$

Therefore $\mathbf{\frac{\sin \theta}{\theta} \ge \frac{2}{\pi}}$ in the given interval.

Some Useful Links:

Our ISI CMI Program

How to use invariance in Combinatorics – ISI Entrance Problem – Video

One comment on “An inequality related to (sin x)/x function | ISI BMath 2007”

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.