Prove that $\sin(x + y + z) \ge \frac45$ for reals with $\sum\limits_{\mathrm{cyc}} \sin x = 2$ and $\sum\limits_{\mathrm{cyc}} \cos x = \frac{11}{5}$

Asked Aug 25 '23 at 01:24

Active Aug 25 '23 at 01:35

Viewed 113 times

Problem. Let $x, y, z$ be real numbers with $\sin x + \sin y + \sin z = 2$ and $\cos x + \cos y + \cos z = \frac{11}{5}$. Prove that $$\sin(x + y + z) \ge \frac45.$$

This question is related to these two questions Q1, Q2.

The result can be verified by Mathematica.
In detail, denote $c_1 = \cos x, c_2 = \cos y, c_3 = \cos z, s_1 = \sin x, s_2 = \sin y, s_3 = \sin z$.
We need to prove that $s_1c_2c_3 - s_1s_2s_3 + c_1s_2c_3 + c_1c_2s_3 \ge \frac45$
given that $c_1^2 + s_1^2 = c_2^2 + s_2^2 = c_3^2 + s_3^2 = 1$ and $s_1 + s_2 + s_3 = 2$ and $c_1 + c_2 + c_3 = \frac{11}{5}$.
Mathematica says it is true.

Equality holds if e.g. $\cos x = \frac45, \cos y = \frac45, \cos z = \frac35, \sin x = \frac35, \sin y = \frac35, \sin z = \frac45$.

Is there a nice/simple proof?

edited Aug 25 '23 at 01:35

asked Aug 25 '23 at 01:24

River Li

49,125

Is there a reason for including optimization and maxima-minima as tags? – Tony Mathew Aug 25 '23 at 02:04
@TonyMathew The minimum of $\sin (x + y + z)$ is $4/5$. It is an optimization problem as well. – River Li Aug 25 '23 at 02:10
It has been quite many weeks, has no progress been made on this problem? – Sahaj Nov 05 '23 at 13:09
@SahajSatishSharma Thanks. No simple proof is available. – River Li Nov 05 '23 at 13:43

Prove that $\sin(x + y + z) \ge \frac45$ for reals with $\sum\limits_{\mathrm{cyc}} \sin x = 2$ and $\sum\limits_{\mathrm{cyc}} \cos x = \frac{11}{5}$

0 Answers0

Linked