Mathematics

Problem

(a) On the same set of axes, sketch neat labelled graphs of

y = \sin x

and

y = \cos x

for

0 \le x \le 2\pi

. Mark all intercepts on the

x

–axis and the

y

–axis. (b) Using your sketch, explain why

\cos x

may be viewed as a horizontal shift of the graph of

\sin x

. State the magnitude and direction of this shift in terms of

\pi

. (c) From the sketch, list: (i) the values of

x

[0,2\pi]

for which

\sin x = 0

, and (ii) the values of

x

[0,2\pi]

for which

\cos x = 0

Intuitive explanation

The sine and cosine functions are closely related periodic waves—the key insight is that cosine is simply sine shifted horizontally. By sketching both curves carefully on the same axes and marking where each crosses the axes, you'll see the visual pattern that reveals their phase relationship. This visual approach transforms an abstract algebraic relationship into something you can literally see and measure on the graph, making it easy to identify intercepts and describe the shift precisely.

Intuitive explanation

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