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Let ${t_1},{t_2},{t_3}$ be three distinct points on circle ${\left| t \right| = 1}$ . If ${{\theta _1},{\theta _2},{\theta _3}}$ be the arguments of ${t_1},{t_2},{t_3}$ respectively then range of $$({\cos \left( {{\theta _1} - {\theta _2}} \right) + \cos ({\theta _2} - {\theta _3}) + \cos ({\theta _3} - {\theta _1})})$$

I have tried plotting points and maybe i think if points will be at 120 degrees to each other maybe then this expression will be minimum . But further i don't know how to proceed . Please help . Thank you

Diglett
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klaus_03
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1 Answers1

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Let $\theta_1-\theta_2=2x$ etc. so that $x+y+z=0$

$$P=\cos2x+\cos2y+\cos2z=2\cos(x+y)\cos(x-y)+2\cos^2z-1$$

Replace $x+y=-z$ to form a quadratic equation in $\cos z$

Now follow

$ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $

In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$

In a triangle, find the minimum and maximum of $\cos(A-B)\cos(B-C)\cos(C-A)$

lab bhattacharjee
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