Periodic Function in Trigonometry

Periodic Function in Trigonometry:

A function f(x) is said to be periodic if there exists a real number λ > 0 such that f(x + λ) = f(x) for all x.

The smallest positive value of λ for which f(x + λ) = f(x) is known as the period of the function f(x).

We have sin (2π + θ) = sin θ, cos (2π + θ) = cos θ, cosec (2π + θ) = cosec θ and sec (2π + θ) = sec θ. Also, tan (π + θ) = tan θ and cot (π + θ) = cot θ. Thu, trigonometric functions are all periodic. The periods of sine, cosine, cosecant and secant function are 2π, whereas the period so tangent and cotangent functions are π. We consider the functions cos 3θ.

One can write cos 3θ = cos (3θ + 2π) = cos 3(θ +2π/3). Thus, when θ is replaced by θ +2π/3, the value of the function remains unchanged. As 2π/3 is the least positive constant for which the value of cos 3θ remains unchanged, the period of cos 3θ is 2π/3.