A smooth periodic curve passes through the origin

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

Problem

O(0,0)

. Moving in the positive

x

-direction, the curve attains its first maximum value

1

x=\tfrac{\pi}{2}

and its next minimum value

-1

x=\tfrac{3\pi}{2}

. Thereafter the pattern repeats indefinitely in both directions. (i) Prove that the curve satisfies the equation

y=\sin x

for all real

x

. (ii) Write a general expression for all

x

-coordinates at which the curve attains its maxima and another for all

x

-coordinates at which it attains its minima. (iii) Determine the complete set of zeros of the curve.