Question

Let $z_1,z_{2}and{z}_3$ be three points on $\left|z\right|=1.$ If ${\theta }_1,{\theta }_2$ and ${\theta }_3$ be the arguments of $z_1,z_2,z_3$ respectively, then cos$({\theta }_1-{\theta }_2)$ + cos $({\theta }_2-{\theta }_3)+\cos ({\theta }_3-{\theta }_1)$

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

We have $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$
Now, $|z_1+z_2+z_3|\ge 0$
$⟹{|z_1+z_2+z_3|}^2\ge 0$
$⟹{\left|z_1\right|}^2+{\left|z_2\right|}^2+{\left|z_3\right|}^2+2Re(z_1{\overline{z}}_2+z_2{\overline{z}}_3+{\overline{z}}_3{\overline{z}}_1)$ $\ge$ 0
$⟹3+2\left\{\cos ({\theta }_1-{\theta }_2)+\cos ({\theta }_2-{\theta }_3)+\cos ({\theta }_3-{\theta }_1)\right\}\ge 0$
$⟹\cos ({\theta }_1-{\theta }_2)+\cos ({\theta }_2-{\theta }_3)+\cos ({\theta }_3-{\theta }_2)$ $\ge -\frac{3}{2}$