Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

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 $\mathbf{\sin \theta \geq \frac{2\theta}{\pi}}$.

Discussion:

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

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com