Precalculus

Trigonometry

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Periodicity of Trigonometric Functions

Periodic Processes and Functions

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

The Moon revolving around the Earth.
Figure 1.

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

Definition of a periodic function

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.

Proof of sine and cosine periodicity.
Figure 2.

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

Periodicity of cosine
Periodicity of sine

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 \(\pi:\)

Periodicity of tangent

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 \(\pi:\)

Periodicity of cotangent

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 \(2\pi:\)

Periodicity of secant

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 \(2\pi:\)

Periodicity of cosecant

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

See solved problems on Page 2.

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