# Angles in a triangle (Tomato Subjective 116)

Problem: If A, B, C are the angles of a triangle, then show that $$\displaystyle { \sin A + \sin B – \cos C \le \frac {3 \sqrt{3}}{2}}$$

Discussion:

$$\displaystyle { \sin A + \sin B – \cos C }$$
$$\displaystyle { = \sin A + \sin B – \cos (\pi – (A+B)) }$$
$$\displaystyle { = \sin A + \sin B + \sin (A+B) }$$
$$\displaystyle { = 2\sin \frac{(A+B)}{2} \cos \frac{(A-B)}{2} + 2\sin \frac{(A+B)}{2} \cos \frac{(A+B)}{2} }$$
$$\displaystyle { = 2\sin \frac{(A+B)}{2} \left( \cos \frac{(A-B)}{2} + \cos \frac{(A+B)}{2}\right ) }$$
$$\displaystyle { = 2\sin \frac{(A+B)}{2} 2\cos \frac{A}{2} \cos \frac{B}{2} }$$
$$\displaystyle { = 4\sin \frac{(\pi -C)}{2} \cos \frac{A}{2} \cos \frac{B}{2} }$$
$$\displaystyle { = 4\sin \left(\frac{\pi}{2} – \frac{C}{2} \right) \cos \frac{A}{2} \cos \frac{B}{2} }$$
$$\displaystyle { = 4\cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} }$$

We apply Jensen’s Inequality and Arithmetic Mean – Geometric Mean inequality here. Since cosine function is concave in the interval $$[0, \frac{\pi}{2} ]$$, we have
$$\displaystyle { \left (\cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} \right )^{\frac{1}{3}} \le \frac{\cos \frac{C}{2} + \cos \frac{A}{2} +\cos \frac{B}{2}}{3} \le \cos \left ( \frac{1}{3}\times \frac{A}{2} + \frac{1}{3}\times \frac{B}{2} + \frac{1}{3}\times \frac{C}{2} \right ) }$$
This implies $$\displaystyle { \left (\cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} \right )^{\frac{1}{3}} \le \cos \left ( \frac{A+B+C}{6}\right ) }$$
$$\displaystyle { \Rightarrow \left (\cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} \right )^{\frac{1}{3}} \le \cos \frac{\pi}{6} }$$
$$\displaystyle { \Rightarrow \left (\cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} \right )^{\frac{1}{3}} \le \frac{\sqrt{3}}{2} }$$
$$\displaystyle { \Rightarrow \cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} \le \frac{3\sqrt{3}}{8} }$$
$$\displaystyle { \Rightarrow 4\cos \frac{C}{2} \cos \frac{A}{2} \cos \frac{B}{2} \le \frac{3\sqrt{3}}{2} }$$

## Chatuspathi:

• What is this topic: Properties of triangle
• What are some of the associated concept: Jensen Inequality, Arithmetic Mean-Geometric Mean Inequality
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, discusses these topics in the ‘Inequality’ module.
• Book Suggestions: ‘Secrets in Inequality’ by Pam Kim Hung