Two curves are given by and

MATHEMATICS · Subjective · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Two curves are given by

y_{1}=\sin x

and

y_{2}=\cos x

. (a) Sketch both curves on the same pair of axes for

0\le x\le 2\pi

, clearly labelling each curve. (b) Using the sketch, show that the equation

\sin x = \cos x

has exactly two distinct solutions in

[0,2\pi]

, and write those solutions. (c) Prove, with the aid of the sketches, that between any two consecutive maxima of

y=\sin x

there exists exactly one value of

x

at which

\sin x = \cos x

. Give a clear geometrical argument based on the behaviour of the two graphs.