Examples

Example:

1. sin x = sin (x + 2л) = sin (x + 4л)= …

So sinx is a periodic function with the period 2л. This is alsoncalled Sinusoidal periodic function.

2. The trignometric functions sinx and cosx are periodic functions with functions fundamental (primitive) period 2л.

3. sin 2x and cos 2x are also periodic functions with fundamental period л.

4. tanx is a periodic function with period л.

5. Find the period of sin nx where n is a positive integer shoqmi od

Solution: Let f (x) = sin nx = sin (nx + 2л)

Therefore, 2л /n is the period of sin nx, 2π/n is the period of coxnx π/n is the period of cos nx

6. Show that a constant has any positive number as period.

Solution :

Let f(x) = c, where is a constant.

Then f(x + k) = c, k being any positive number

that is f(x + k) = f(x)

So f(x) is periodic with period k.

Note :

Since, there is no least value of k, we say that f(x) = c has no fundamental period.

7. Let f : RR be the function defined by

Let p be any rational number. If x is rational, then x + p is also rational and if x is irrational, then x + p is also irrational.

Hence, every rational number is a period of f and ƒ has no fundamental period.

8. Let ƒ and g be periodic functions with period p and let a and b be real numbers. Prove that af + bg is also a periodic function with period p.

Solution: Since, ƒ and g are periodic with period p

Hence, af + bg is periodic with period p.

9. If p is a period of f (x), show that np is also a period where n is any integer (positive or negative)

Solution: Since