A horizontal translation of the graph of by units results in a new graph that exactly…

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

Problem

A horizontal translation of the graph of

y=\sin x

k

units results in a new graph that exactly coincides with the graph of

y=\cos x

. Answer the following. (i) Determine the value of

k

in the interval

0<k<2\pi

and specify whether the translation is to the left or to the right. (ii) Verify your answer to part (i) by substituting

x=\dfrac{\pi}{3}

into the two functions and confirming that the ordinates are equal after the translation. (iii) After applying the translation, what happens to the

x

-coordinates of the zeros of

y=\sin x

? Illustrate your answer by listing the first three positive zeros before and after the translation.