Question

If $\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3=\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3=0$, then the value of $\cos \left({\theta }_1+{\theta }_2\right)+\cos \left({\theta }_2+{\theta }_3\right)+\cos \left({\theta }_3+{\theta }_1\right)$ is

• A. $-\frac{1}{2}$
• B. $\frac{1}{2}$
• C. $0$
• D. $1$

Answer 1

Answer: (C)

$z_1=\cos {\theta }_1+i\sin {\theta }_1=e^{i{\theta }_1}$
$z_2=\cos {\theta }_2+i\sin {\theta }_2=e^{i{\theta }_2}$
$z_3=\cos {\theta }_3+i\sin {\theta }_3=e^{i{\theta }_3}$
$z_1+z_2+z_3=0$
$\Rightarrow \overline{z_1}+\overline{z_2}+\overline{z_3}=0$
$\Rightarrow \frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}=0$
$\therefore z_1{z}_2+z_2{z}_3+z_3{z}_1=0$
$\Rightarrow e^{i\left({\theta }_1+{\theta }_2\right)}+e^{i\left({\theta }_2+{\theta }_3\right)}+e^{i\left({\theta }_3+{\theta }_1\right)}=0$
$\therefore \Sigma \cos \left({\theta }_1+{\theta }_2\right)=0$