# Trigonometric inequality $|\sin{a_1}|+|\sin{a_2}|+…+|\sin{a_n}|+|\cos{(a_1+a_2+…+a_n)}| \ge1$ for all real $a_i$

Prove that for all real numbers $a_1,a_2,...,a_n$ the following inequality holds: $$|\sin{a_1}|+|\sin{a_2}|+...+|\sin{a_n}|+|\cos{(a_1+a_2+...+a_n)}| \ge 1$$

• The only thing I tried is the normal mathematical induction, but it doesn't help me – HeatTheIce Jan 3 '15 at 18:51
• Too bad, because that was what I was going to suggest. Does the sum of angles forumula for $\cos()$ help us at all? Maybe something with that and the triangle inequality. – Mike Pierce Jan 3 '15 at 18:54

HINT: $\sin^2x+\cos^2x=1$, $\sqrt{a^2+b^2}\leq|a|+|b|$, $|c+d|\leq|c|+|d|$.

• That helped me to solve it without induction, thank you. – HeatTheIce Jan 3 '15 at 19:10
• @SoulEater You are welcome. – Przemysław Scherwentke Jan 3 '15 at 19:11

HINT: $$\sin(a_1+a_2)=\sin(a_1)\cos(a_2)+\sin(a_2)\cos(a_1),$$ so $$|\sin(a_1+a_2)|\le|\sin(a_1)|+|\sin(a_2)|.$$ Now you can use induction.