# 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

- 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 - Draw a graph of sin and cosec in the same diagram.
- 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 | - Graphically solve the equation 3 cos x +2 = 0, where 0 < x < .
- 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

**1.**(i) range is -1 to 1, amplitude is 1, period is 2/3 (i.e. 120°)

(ii) range is - 3 to 3, amplitude is 3, period is 2 (i.e. 360°)

(iii) range is -0·3 to 0·3, amplitude is 0·3, period is 2/3 (i.e. 120°)

**4.**x = 2·30 (radians)

**5.**x = 0·77 (radians)