Periodic function in Trigonometry

A function f(x) is said to be periodic function in trigonometry if there exists a real number T>0 such that f(x + T) = f(x) for all x.
If T is the smallest positive real number such that f(x+T) = f(x) for all x, then T is called the fundamental period of f(x).
A pattern in which y -values repeats at regular interval. One such a complete pattern is called Cycle. It may begin at any point on the graph. A period of the function is the horizontal shift in the cycle.
Since sin(2nπ + θ) = sin θ, cos(2nπ + θ) = cos θ, for all values of θ and n $\epsilon$ N.
Therefore, sine and cosine functions are periodic functions.
We find that 2π is the smallest positive real number such that
sin (2π + θ) = sin θ and cos(2π + θ) = cos θ for all values of θ.
So sine and cosine function are periodic function with period 2π.
We also know that tan( π + θ ) = tan θ and cot ( π + θ ) = cot θ
Therefore tan θ and cot θ are periodic functions with period π.
 Sine function : Period 2π cosine function : Period 2π tangent function : Period π cosecant function : Period 2π secant function : Period 2π cotangent function : Period π

Examples on Periodic function in Trigonometry

1) Find the period of the function f(x) = -2 cos(3x) .
Solution : The function f(x) = -2 cos(3x) runs through a full cycle when the angle 3x runs from 0 to 2π .
So period of the given function is
3x = 2π
∴ x = $\frac{2π}{3}$
so the period of f(x) = $\frac{2π}{3}$

2) Find the period of the function f(x) = 3 sin (4x).
Solution : The function f(x) = 3 sin (4x) runs through a full cycle when the angle 4x runs from 0 to 2π .
So period of the given function is
4x = 2π
∴ x = $\frac{π}{2}$
so the period of f(x) = $\frac{π}{2}$

3) Find the period of the function f(x) = -2 cos$\frac{θ}{6}$ .
Solution : The function f(x) = -2 cos$\frac{θ}{6}$ runs through a full cycle when the angle $\frac{θ}{6}$ runs from 0 to 2π .
So period of the given function is
$\frac{θ}{6}$ = 2π
∴ θ = 12π