Question

If $\cos \alpha +\cos \beta +\cos \gamma =0$, where $0<\alpha \le \frac{\pi }{2},0<\beta \le \frac{\pi }{2},0<\gamma \le \frac{\pi }{2}$, then what is the value of $\sin \alpha +\sin \beta +\sin \gamma$

• A. $0$
• B. $3$
• C. $\frac{5\sqrt{2}}{2}$
• D. $\frac{3\sqrt{2}}{2}$

Answer 1

The correct option is B $3$

$\cos \alpha +\cos \beta +\cos \gamma =00<\alpha \le \frac{\pi }{2}$

$0<\beta \le \frac{\pi }{2}$

$0\le \gamma \le \frac{\pi }{2}$

In $(0,\frac{\pi }{2}]\to \cos x$ is always positive

$\therefore$ For sum of three cosines to be zero in range $(0,\frac{\pi }{2}]$, all cosine should become individually zero

$\therefore \alpha =\beta =\gamma =\frac{\pi }{2}$

$\therefore \sin \alpha +\sin \beta +\sin \gamma =3$.