Question

The value of the expression $\sin x+\sin y+\sin z$ where $x,y,z$ are real numbers satisfying $x+y+z={180}^{\circ }$ is

• A. Positive
• B. Zero
• C. $-3$
• D. Negative

Answer 1

The correct option is A Positive

When $x+y+z={180}^0$,

$\sin x+\sin y+\sin z=4\cos \frac{x}{2}\cos \frac{y}{2}\cos \frac{z}{2}$

As, $\frac{x}{2}+\frac{y}{2}+\frac{z}{2}={90}^0$,

$\frac{x}{2},\frac{y}{2},\frac{z}{2}$ are acute angles.

So, $4\cos \frac{x}{2}\cos \frac{y}{2}\cos \frac{z}{2}>0$

ie, $\sin x+\sin y+\sin z>0$

Option A is correct answer.