For real numbers and

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

Problem

For real numbers

\alpha

and

\beta

, let

P(\cos\alpha,\sin\alpha)

and

Q(\cos\beta,\sin\beta)

be two points on the unit circle with centre at the origin. (i) Prove that the length of the chord

PQ

2\sin\dfrac{|\alpha-\beta|}{2}

. (ii) Using part (i), find exactly the length of the chord that subtends an angle of

20^\circ

at the centre of the unit circle. (iii) Discuss briefly why the expression obtained in part (i) depends only on the absolute difference

|\alpha-\beta|

and not on the individual angles.