On the same set of axes

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

Problem

On the same set of axes, study the functions

y=\sin x

and

y=\cos x

for one complete period. (i) Draw the graphs of both functions for

0 \le x \le 2\pi

. Label the key points on each curve. (ii) Determine analytically all values of

x

0 \le x \le 2\pi

for which

\sin x = \cos x

, and show these points on the sketch. (iii) Show that the horizontal distance between two successive maxima of

\cos x

and the nearest maxima of

\sin x

\frac{\pi}{2}

. Illustrate this clearly on the diagram. (iv) Using the plotted curves, explain why the relationship

\cos x = \sin\left(\frac{\pi}{2}-x\right)

holds for all real

x