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Sketch the graphs of the following pairs of functions on the same axes:

(i) f (x) = sin 2x, g (x) = 2 sin x

(ii) f (x) = sin x/2, g (x) = sin x

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(i) f (x) = sin 2x, g (x) = 2 sin x

As we know that the functions f(x) = sin 2x and g (x) = 2 sin x are periodic functions with periods π and π.

Here, the values of these functions are tabulated below:

The values of f (x) = sin (2x) in [0, π]

x 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4
f (x) = sin (2x) 0 1 0 -1 0 1 0 -1 0

The values of g (x) = 2 sin x in [0, π]

x 0 π/2 π 3π/2
g (x) = 2 sin x 0 1 0 -1 0

Thus, the required curve is:

(ii) f (x) = sin x/2, g (x) = sin x

As we know that the functions f(x) = sin x/2 and g (x) = sin x are periodic functions with periods π and 2π.

Here, the values of these functions are tabulated below:

The values of f (x) = sin x/2 in [0, π]

x 0 π
f (x) = sin x/2 0 1 0 -1 0

The values of g (x) = sin (x) in [0, 2π]

x 0 π/2 π 3π/2 5π/2 7π/2
g (x) = sin (x) 0 1 0 -1 0 1 0 -1 0

Thus, the required curve is

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