Periodicity of Circular Functions

A function f is said to be periodic if there exists a constant real quantity p such that
f (x +p) = f (x) for all x D_f

There may exist more than one value of p satisfying the above relation. The least positive value of p satisfying above relation is called the period of f.

We know that sin (x + p) = sin x for all real x, where p = ± 2 , ± 4, ± 6, ...
In general, sin (x +2 n ) = sin x for all real x and n Z
Thus the period of sin x (or sin ) is 2 (or 360°).

The period of cos, sec, cosec is also 2 (or 360°).

However for tangent and cotangent functions, we have
tan (x + n) = tan x,
cot (x + n) = cot x.
So the period of tan and cot is (or 180°).

In general, period of [a sin (bx +c)] or [a cos (bx +c)] is 2/| b |

Exercise

1. Draw the graphs of following functions. Also mention their range, amplitude and period of cycle.
   (i) sin 3 x
   (ii) 3 sin x
   (iii) 0·3 sin 3 x
2. Draw a graph of sin and cosec in the same diagram.
3. Draw the graphs of the following:
   (i) cos (x -/2)
   (ii) cos (x -/4)
   (iii) cos (x +/4)
   (iv) 3 + 2 cos (2x - /6)
   (v) cos x -sin x
   (vi) sin²x
   (vii) | sin x |
4. Graphically solve the equation 3 cos x +2 = 0, where 0 < x < .
5. Draw the graph of y = cos 2 x +cos x for values of x from 0 to . On the same diagram, draw the graph of y = x. Hence estimate the positive root of the equation x = cos 2 x +cos x.

Answers