Consider two rays and in the plane

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Consider two rays

OP

and

OQ

in the plane, where

P

and

Q

lie on the unit circle and the angles they make with the positive

x

–axis are

x

and

y

respectively. (i) Define the unit vector corresponding to an angle

\theta

in standard position. (ii) Express the dot product of the unit vectors along

OP

and

OQ

in terms of

x

and

y

by writing their components explicitly. (iii) Hence derive a formula for

\cos(x-y)

involving

\cos x,\cos y,\sin x

and

\sin y

. State each step clearly.