Graphs of Trigonometric Functions

Sine, cosine, and tangent are the most important trigonometric functions. To obtain the graphs of trigonometric functions we need to know the period, phase, and amplitude. In this article, we will learn about the graphical representations of the sine, cosine, and tangent functions.

We will also learn about the amplitude, phase, and period of these functions.

ALGEBRA
graphs of trigonometric functions

Relevant for

Learning to graph fundamental trigonometric functions.

See graphs

ALGEBRA
graphs of trigonometric functions

Relevant for

Learning to graph fundamental trigonometric functions.

See graphs

Graph of sine function

The sine function is one of the fundamental trigonometric functions. We can write y=\sin(x) or f(x)=\sin(x). The following is the graph of the sine:

graph of sine function

The sine function has the following characteristics:

  • The roots or zeros of y=\sin(x) are multiples of π.
  • The graph of sine passes through the origin since when x is 0, we have \sin(0) = 0.
  • The period of sine function is 2π.
  • The curve height at each point is equal to the value of the sine function at each point.
  • The maximum value of the function is 1. This happens when x=\frac{\pi}{2}.
  • The minimum value of the function is -1. This happens when x=\frac{3\pi}{2}.

Graph of cosine function

For the cosine function, we can write y=\cos(x) or f(x)=\cos(x). The following is the graph of cosine:

graph of cosine function

The following are the characteristics of the graph of the cosine function:

  • We have this identity \sin(x+\frac{\pi} {2}) = \cos(x ).
  • The graph of the cosine function is the graph that we obtain if we translate the graph of \sin\frac{\pi} {2} units to the left.
  • The roots or zeros of the cosine function are the multiples of \frac{\pi} {2}.
  • The graph of the cosine function crosses the y-axis at the point (0, 1).
  • The period of the cosine function is 2π.
  • The maximum value of the function is 1. This happens when x=0.
  • The minimum value of the function is -1. This happens when x is π.

We can observe the following similarities between sine graphs and cosine graphs:

  • They both produce the same curve, which is translated along the x-axis.
  • They both have an amplitude of 1.
  • They both have a period of 2π or 360°.

The following graph shows us a comparison of the sine and cosine functions:

graph of cosine and sine functions

Graph of tangent function

The graph of the tangent is completely different compared to the graph of the sine and the cosine functions. The graph of the tangent function has y values ranging from negative infinity to positive infinity and crosses the x-axis over a period of π radians:

graph of tangent function

The following are the characteristics of the tangent function:

  • The roots or zeros of the function are the multiples of π.
  • The graph of the tangent passes through the origin.
  • The graph of the tangent has an indefinite amplitude because the curve goes to infinity.
  • The period of the sine function is π or 180°.

Amplitude of trigonometric functions

The amplitude of trigonometric functions is the distance from the centerline of a trigonometric function to its maximum or minimum point. The amplitude is the absolute value by which a trigonometric function is multiplied.

The amplitude of a function can also be measured by measuring the distance of the highest point from the lowest point and dividing by 2. The amplitude tells us how high or low the graph of a trigonometric function is.

amplitud of trigonometric functions

Period of trigonometric functions

The period is the measure from one point on the graph, for example, a maximum point, to the next equivalent point. The following is a graphical representation of the period of the sine function:

period of trigonometric functions

Phase of trigonometric functions

The phase is a measure of the horizontal translation of a trigonometric function from its original position. The following is a graphical representation of the phase of the sine function:

phase of trigonometric functions

How to graph trigonometric functions?

There are several methods that can be used to graph trigonometric functions. The following is a detailed explanation of one of the most efficient methods for graphing trigonometric functions.

To plot the graph of trigonometric functions, we have to start by converting the given function to the general form f(x)=a\sin(bx-c) + d. In this way, we can easily find the different parameters such as amplitude, phase, period, and vertical translation, where:

  • |a|= amplitude
  • \frac{{2\pi}}{|b|}= period
  • \frac{{c}}{b}= phase
  • d= vertical translation

Similarly, we can find the graph of the cosine function using the form f(x)=a\cos(bx-c)+d.

The graphs of the fundamental trigonometric functions are shown below in their base form:

graph of basic sine, cosine and tangent functions

EXAMPLE

Sketch the graph of the function y=3\sin({2x})+4.

Solution:

  • The amplitude is 3, so the distance between the minimum and maximum values is 6.
  • The number of waves is 2. Each wave has a period of 360°÷2=180°.
  • The graph is translated up by 4 units.
  • The maximum point is (3×1)+4=7 and the minimum point is (3×-1)+4=1.
  • The period is \frac{{2\pi}}{2}=\pi

The graph looks like this:

graph of trigonometric function with parameters

See also

Interested in learning more about graphs of functions? Take a look at these pages:

Learn mathematics with our additional resources in different topics

LEARN MORE