Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A, B, C$ be angles in a triangle. Is the following inequality $$4\cos A \le 1 + \cos\left(\frac{B-C}{2}\right)$$ true? I just assume it but don't have a proof. Thank you for your help.

share|improve this question
    
Thank you N.S. This was very stupid attempt to prove the true inequality $$4(\cos A + \cos B + \cos C) \le 3 + \cos(\frac{B-C}{2}) + \cos(\frac{C-A}{2}) + \cos(\frac{A-B}{2}).$$ Any idea how this can be done? –  Martin Jul 23 '12 at 15:34
add comment

1 Answer 1

Nope it is not true.

The right side is at most 2, while when $A$ is very small the left side is close to 4.

Note that if $0 <A< 30^\circ$ then

$$4 \cos(A) > 2 \geq 1 + \cos\left(\frac{B-C}{2}\right)$$

share|improve this answer
    
Thank you. This was very stupid attempt to prove the true inequality $$4(\cos A + \cos B + \cos C) \le 3 + \cos(\frac{B-C}{2}) + \cos(\frac{C-A}{2}) + \cos(\frac{C-A}{2}).$$ Any idea how this can be done? –  Martin Jul 23 '12 at 15:24
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.