Periods of Trigonometric Functions

Interactive tutorial to explore the periods of trigonometric functions

Controls

Select Trigonometric Function

Period of Selected Function

0π (0)
f(x) = f(x + P)

Note: Functions with vertical asymptotes (tan, cot, sec, csc) show discontinuities where the function is undefined.

Original function f(x)
Shifted function f(x + P)

Graph Visualization

Adjust Period Shift (P):
0π (0)

Drag the slider to shift the function and find where f(x) = f(x + P)

Your browser does not support the HTML5 canvas element. Please use a modern browser to view this interactive graph.

sin(x): The sine function has a period of 2π. This means sin(x) = sin(x + 2π) for all x.

Interactive Tutorial Instructions

  1. Select one of the 6 trigonometric functions using the function buttons.
  2. Use the slider (above the graph) to adjust the period shift (P), starting from P = 0.
  3. Slowly increase P until the blue graph (original function) and red graph (shifted function) are identical (superimposed).
  4. The value of P displayed as a multiple of π is the period of the selected function.
  5. Select another trigonometric function and repeat the exploration.

Understanding Periods of Trigonometric Functions

A function f is periodic if there exists a positive number P such that:

f(x) = f(x + P) for all x in the domain of f

The smallest such P is called the period of the function.

This interactive tool lets you explore the periods of the six basic trigonometric functions. Use the slider to shift the function and find the smallest P where the function repeats itself.

More on Trigonometric Functions

sin(x) cos(x) tan(x) cot(x) sec(x) csc(x)

Before starting this tutorial, you might want to review periodic functions.