Let and be the points on the unit circle corresponding to the angles and respectively

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Let

Q_1,\,Q_2

and

Q_3

be the points on the unit circle corresponding to the angles

\theta,\;\theta+\dfrac{2\pi}{3}

and

\;\theta+\dfrac{4\pi}{3}

respectively, where

\theta

is any real number. (i) Write the coordinates of

Q_1,\,Q_2

and

Q_3

in terms of

\cos

and

\sin

. (ii) Show that the distances

Q_1Q_2,\,Q_2Q_3

and

Q_3Q_1

are all equal. (iii) Hence prove that

Q_1Q_2Q_3

is an equilateral triangle and find the exact value of its side length.