Question

Determine whether the given function is periodic. If so, find its fundamental period. $$ \tan \pi x $$

Short answer

If yes, what is its fundamental period? Answer: Yes, the function \(\tan(\pi x)\) is periodic. Its fundamental period is 1.

Step-by-step solution

Identify the given function and periodic properties of tangent function

The given function is \(\tan(\pi x)\). The tangent function has a period of \(\pi\), which means that \(\tan(x) = \tan(x+k\pi)\) for integer values of k.

Check if the given function repeats after a certain period

We want to find a constant P such that \(\tan(\pi x) = \tan(\pi (x + P))\). Using the periodic property of the tangent function, we have \(\tan(\pi x) = \tan(\pi x + k\pi)\) for integer values of k. Comparing the above equation with \(\tan(\pi x) = \tan(\pi (x + P))\), we get \(\pi (x + P) = \pi x + k\pi\). Then, we simplify the equation by dividing by \(\pi\).

Simplify the equation and solve for P

Dividing by \(\pi\), we get \(x + P = x + k\), which simplifies to \(P = k\). Since P is a constant, we can choose the smallest positive integer value for k, which is 1. Therefore, the fundamental period of the given function is \(P = 1\).

Conclusion

The function \(\tan(\pi x)\) is periodic with a fundamental period of 1.

Key concepts

Tangent Function

The tangent function is a fundamental trigonometric function that is widely used in mathematics, particularly within the field of trigonometry.

The function is defined as the ratio of the sine and cosine functions, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). The tangent function is unique among the trigonometric functions because it is not bounded between -1 and 1, but can take any real value. This is due to the fact that as the cosine function approaches zero, the value of the tangent function increases without any limit.

When graphed, the tangent function exhibits a repeating pattern, with asymptotes (lines that the graph approaches but never touches) where the function is undefined, which occurs at \( \frac{\pi}{2} \) plus any integer multiple of \( \pi \). These asymptotes correspond to the angles where the cosine function equals zero, and the sine function has a non-zero value.

The tangent function illustrates an important property in trigonometry:

• It is periodical, repeating its values in regular intervals.
• The standard period of the tangent function is \( \pi \), meaning that \( \tan(x) = \tan(x + k\pi) \) for any integer \( k \).

This periodic nature of the tangent function is crucial when solving trigonometric equations and modeling periodic phenomena.

Fundamental Period

Understanding the concept of the fundamental period is crucial when studying periodic functions in trigonometry.

The fundamental period of a periodic function is the smallest positive interval over which the function completes one full cycle and begins to repeat itself. Each periodic function has a characteristic fundamental period which depends on its specific properties.

In the context of trigonometric functions, each has its own fundamental period:

• The sine and cosine functions have a fundamental period of \( 2\pi \).
• The tangent function, as mentioned previously, has a shorter fundamental period of \( \pi \).

Knowing the fundamental period allows us to understand the function's behavior and predict its values at any given point.

For example, if we consider the function \( \tan(\pi x) \) from the exercise, by applying the property that the tangent function repeats every \( \pi \) units, we can deduce that this scaled function will have a different fundamental period. Instead of repeating every \( \pi \) like the standard tangent function, the function \( \tan(\pi x) \) will repeat every 1 unit on the \( x \) axis, as the \(\pi\) inside the function effectively 'scales' its periodicity.

Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles, more specifically right-angled triangles. Derived from the Greek words 'trigonon' (triangle) and 'metron' (measure), it literally means 'measuring the sides of triangles'.

The fundamental trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each of which relates a specific angle to a ratio of two sides of a right triangle. These functions are essential tools for understanding and describing the geometric properties of circles and wave-like phenomena.

Trigonometry has applications across various fields such as physics, engineering, astronomy, and even in everyday life where wave patterns, cycles, and oscillations are present. It provides the mathematical underpinnings for dealing with periodic processes, such as sound waves, light waves, and the ebb and flow of tides.

The subject also delves into more complex functions and their properties, including the periodic nature of the functions and how to manipulate and transform these functions to suit specific problem-solving scenarios. Such an understanding is displayed in the exercise where the properties of the tangent function's periodicity were utilized to determine the fundamental period of \( \tan(\pi x) \).