# Periodicity of Trigonometric Functions

## Periodic Processes and Functions

We often encounter periodic phenomena in the nature, technology, and human society. Recall the 24-hour day-night cycle, or tidal cycles caused by the Moon revolving around the Earth.

Another example is a pendulum. When a pendulum makes one complete swing over and back in $$T$$ seconds, the deflection of the pendulum from its equilibrium position will be the same at times $$t,$$ $$t + T,$$ $$t + 2T,$$ etc.

Periodic processes are described using periodic functions.

A positive real number $$T$$ is called the period of a function $$f$$ if

$f\left({t}\right) = f\left({t + T}\right)$

for all values of $$t$$ from the domain of $$f.$$

If $$T$$ is a period of a function $$f,$$ then the product $$nT,$$ where $$n \in \mathbb{Z},$$ is also a period of the function $$f:$$

$f\left( t \right) = f\left( {t + nT} \right).$

In particular for $$n = -1,$$ we have

$f\left( {t - T} \right) = f\left( t \right).$

The least positive period of a function is called the fundamental period or simply the period of the function.

## Period of Sine and Cosine Functions

The sine and cosine functions are periodic, with period $$2\pi.$$

Indeed, consider two points $$M\left( \theta \right)$$ and $$N\left( {\theta + 2\pi } \right)$$ lying on the unit circle.

These points coincide and have the same coordinates. Since the point $$M\left( \theta \right)$$ has coordinates $$\cos\theta$$ and $$\sin\theta,$$ we can write

$\cos\theta = \cos\left({\theta + 2\pi}\right)$
$\sin\theta = \sin\left({\theta + 2\pi}\right)$

These relationships show that $$2\pi$$ is one of the periods of sine and cosine.

Prove that $$2\pi$$ is the least period for these functions. By contradiction, suppose there is a period $$T$$ less than $$2\pi$$ for the cosine function. Then we have

$\cos \left( {\theta + T} \right) = \cos \theta .$

This identity is valid for any $$\theta,$$ so let $$\theta = 0:$$

$\cos T = \cos 0 = 1.$

The equation $$\cos T = 1$$ has the following solutions: $$T = 0, 2\pi, 4\pi, 6\pi, \ldots$$ However by our assumption, $$0 \lt T \lt 2\pi.$$ We have here a contradiction. Hence, the equation $$\cos T = 1$$ is false, and the cosine function does not have positive periods less than $$2\pi.$$

The proof for the sine function is carried out in the same way.

## Period of Other Trigonometric Functions

The tangent function has a period of $$\pi:$$

$\tan\theta = \tan\left({\theta + \pi}\right)$

The tangent function is defined for any angles $$\theta$$ except the values where $$\cos \theta = 0,$$ that is, the values $$\frac{\pi }{2} + \pi n,$$ $$n \in \mathbb{Z}.$$

Similarly, the period of the cotangent function is also $$\pi:$$

$\cot\theta = \cot\left({\theta + \pi}\right)$

The cotangent function is the quotient of cosine and sine. Its domain contains all angles $$\theta$$ except the points $$\pi n, n \in \mathbb{Z},$$ where the sine function is equal to zero.

The secant and cosecant are the reciprocal functions of cosine and sine, respectively. Therefore, the secant function is periodic, with period $$2\pi:$$

$\sec\theta = \sec\left({\theta + 2\pi}\right)$

It is defined for all real numbers $$\theta$$ except the points $$\frac{\pi }{2} + \pi n,$$ $$n \in \mathbb{Z}.$$

Cosecant also has a period of $$2\pi:$$

$\csc\theta = \csc\left({\theta + 2\pi}\right)$

Cosecant is defined for all values of $$\theta$$ except $$\pi n, n \in \mathbb{Z}.$$

See solved problems on Page 2.