A point is first rotated anticlockwise about the origin through an angle to and then…

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

A point

P(x, y)

is first rotated anticlockwise about the origin through an angle

\alpha

P_{1}

and then again anticlockwise through an angle

\beta

P_{2}

. (i) Obtain the coordinates of

P_{2}

in terms of

x,\,y,\,\alpha

and

\beta

. Show all necessary steps. (ii) Prove that

P_{2}

coincides with the position obtained by a single anticlockwise rotation of

P

through the angle

(\alpha+\beta)

. (iii) If instead the second rotation were clockwise through the same angle

\alpha

, what would be the coordinates of the final image of

P

? Simplify your answer.