Let , , and be the parametric angles of three points , , and , respectively, lying on…

MATHEMATICS · Single correct · ISC · Mathematics · Sets and Functions · Sets and Functions

Problem

Let

\alpha

\beta

, and

\gamma

be the parametric angles of three points

P

Q

, and

R

, respectively, lying on

x^2+y^2=1

. If the lengths of chords

AP

AQ

, and

AR

are in GP, where

A=(1,0)

and

\alpha,\beta,\gamma\in(0,2\pi)

, then (1)

\sin\frac{\alpha+\gamma}{4}\cos\frac{\alpha-\gamma}{4}\ge\sin\frac{\beta}{2}

(2)

\sin\left(\frac{\alpha+\gamma}{4}\right)\cos\left(\frac{\alpha-\gamma}{4}\right)\le\sin\frac{\beta}{2}

(3)

\sin\frac{\alpha}{2}\sin\frac{\gamma}{2}\ge\sin\frac{\beta}{2}

(4)

\sin\frac{\alpha}{2}\sin\frac{\gamma}{2}\le\sin\frac{\beta}{2}