Question

If ${\sin }^2{\theta }_1+{\sin }^2{\theta }_2+{\sin }^2{\theta }_3=0$, then which of the following is not a possible value of $\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3$

• A. $3$
• B. $-3$
• C. $-1$
• D. $-2$

Answer 1

Answer: (D)

$-2$

${\sin }^2{\theta }_1+{\sin }^2{\theta }_2+{\sin }^2{\theta }_3=0$

$\Rightarrow {\sin }^2{\theta }_1+{\sin }^2{\theta }_2+{\sin }^2{\theta }_3=0$

$\Rightarrow {\cos }^2{\theta }_1,{\cos }^2{\theta }_2,{\cos }^2{\theta }_3=1$

$\Rightarrow \cos {\theta }_1,\cos {\theta }_2,\cos {\theta }_3=\pm 1$

$\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3$ can be

$-3$ (where all are $-1$)

$3$ (when all are $+1$)

$-1$ (when any two are $-1$ and one $+1$)

$1$ (when any two are $+1$ and one $-1$)

but $-2$ is not a possible value.