Graphing Trigonometric Functions

What are the key features of trigonometric functions graphs?

Before we go through the process to graph trigonometric functions, we need to identify some key features about them:

Amplitude

For functions in the form $y=\mathit{a}\sin b\theta$, or $y=\mathit{a}\cos b\theta$, the amplitude equals the absolute value of a.

$Amplitude=\left|a\right|$

The tangent functions graph has no amplitude, as it does not have a minimum or maximum value.

Period

For functions in the form $y=a\sin \mathit{b}\theta$, or $y=a\cos \mathit{b}\theta$, b is known as the horizontal stretch factor, and you can calculate the period as follows:

$Period=\frac{2\mathrm{\pi }}{\left|\mathrm{b}\right|}or\frac{360°}{\left|b\right|}$

For functions in the form $y=a\tan \mathit{b}\theta$, the period is calculated like this:

$Period=\frac{\mathrm{\pi }}{\left|b\right|}or\frac{180°}{\left|b\right|}$

Domain and range

The domain and range of the main trigonometric functions are as follows:

How to graph trigonometric functions?

To graph the trigonometric functions you can follow these steps:

• If the trigonometric function is in the form $y=a\sin b\theta$, $y=a\cos b\theta$, or $y=a\tan b\theta$, then identify the values of a and b, and work out the values of the amplitude and the period as explained above.

• Create a table of ordered pairs for the points that you will include in the graph. The first value in the ordered pairs will correspond to the value of the angle θ, and the values of y will correspond to the value of the trigonometric function for the angle θ, for example, sin θ, so the ordered pair will be (θ, sin θ). The values of θ can be either in degrees or radians.

• Plot a few points on the coordinate plane to complete at least one period of the trigonometric function.

• Connect the points with a smooth and continuous curve.

Sine graph

The graph for a sine function $y=\sin \theta$ looks like this:

Sine graph, Marilú García De Taylor - StudySmarter Originals

From this graph we can observe the key features of the sine function:

• The graph repeats every 2π radians or 360°.

• The minimum value for sine is -1.

• The maximum value for sine is 1.

• This means that the amplitude of the graph is 1 and its period is 2π (or 360°).

• The graph crosses the x-axis at 0 and every π radians before and after that.

• The sine function reaches its maximum value at π/2 and every 2π before and after that.

• The sine function reaches its minimum value at 3π/2 and every 2π before and after that.

Cosine graph

The graph for the cosine function $y=\cos \theta$looks exactly like the sine graph, except that it is shifted to the left by π/2 radians, as shown below.

Cosine graph, Marilú García De Taylor - StudySmarter Originals

By observing this graph, we can determine the key features of the cosine function:

• The graph repeats every 2π radians or 360°.

• The minimum value for cosine is -1.

• The maximum value for cosine is 1.

• This means that the amplitude of the graph is 1 and its period is 2π (or 360°).

• The graph crosses the x-axis at π/2 and every π radians before and after that.

• The cosine function reaches its maximum value at 0 and every 2π before and after that.

• The cosine function reaches its minimum value at π and every 2π before and after that.

Tangent graph

The graph of the tangent function $y=\tan \theta$, however, looks a bit different than the cosine and sine functions. It is not a wave but rather a discontinuous function, with asymptotes:

Tangent graph, Marilú García De Taylor - StudySmarter Originals

By observing this graph, we can determine the key features of the tangent function:

• The graph repeats every π radians or 180°.

• No minimum value.

• No maximum value.

• This means that the tangent function has no amplitude and its period is π (or 180°).

• The graph crosses the x-axis at 0 and every π radians before and after that.

• The tangent graph has asymptotes, which are values where the function is undefined.

• These asymptotes are at π/2 and every π before and after that.

The tangent of an angle can also be found with this formula:

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

What are the graphs of the reciprocal trigonometric functions?

Each trigonometric function has a corresponding reciprocal function:

• Cosecant is the reciprocal of sine.
• Secant is the reciprocal of cosine.
• Cotangent is the reciprocal of tangent.

To graph the reciprocal trigonometric functions you can proceed as follows:

Cosecant graph

The graph of the cosecant function $y=csc\theta$ can be obtained like this:

• Graph the corresponding sine function first, to use it as a guide.
• Draw vertical asymptotes in all the points where the sine function intercepts the x-axis.
• The cosecant graph will touch the sine function at its maximum and minimum value. From those points, draw the reflection of the sine function, which approaches but never touches the vertical asymptotes and extends to positive and negative infinity.

Cosecant graph, Marilú García De Taylor - StudySmarter Originals

Secant graph

To graph the secant function $y=sec\theta$ you can follow the same steps as before, but using the corresponding cosine function as a guide. The secant graph looks like this:

Secant graph, Marilú García De Taylor - StudySmarter Originals

Cotangent graph

The cotangent graph is very similar to the graph of tangent, but instead of being an increasing function, cotangent is a decreasing function. The cotangent graph will have asymptotes in all the points where the tangent function intercepts the x-axis.

Cotangent graph, Marilú García De Taylor - StudySmarter Originals

What are the graphs of the inverse trigonometric functions?

The inverse trigonometric functions refer to the arcsine, arccosine and arctangent functions, which can also be written as $S{\mathrm{in}}^{-1},C{\mathrm{os}}^{-1}$ and $T{\mathrm{an}}^{-1}$. These functions do the opposite of the sine, cosine and tangent functions, which means that they give back an angle when we plug a sin, cos or tan value into them.

However, in order to make the inverses of trigonometric functions become functions, we need to restrict their domain. Otherwise, the inverses are not functions because they do not pass the vertical line test. The values in the restricted domains of the trigonometric functions are known as principal values, and to identify that these functions have a restricted domain, we use capital letters:

Arcsine graph

Arcsine is the inverse of the sine function. The inverse of $y=Sinx$ is defined as $x=Si{n}^{-1}y$ or $x=Arc\sin y$. The domain of the arcsine function will be all real numbers from -1 to 1, and its range is the set of angle measures from $-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$. The graph of the arcsine function looks like this:

Arcsine graph, Marilú García De Taylor - StudySmarter Originals

Arccosine graph

Arccosine is the inverse of the cosine function. The inverse of $y=Cosx$ is defined as $x=Co{s}^{-1}y$ or $x=Arc\cos y$. The domain of the arccosine function will be also all real numbers from -1 to 1, and its range is the set of angle measures from $0\le y\le \mathrm{\pi }$. The graph of the arccosine function is shown below:

Arccosine graph, Marilú García De Taylor - StudySmarter Originals

Arctangent graph

Arctangent is the inverse of the tangent function. The inverse of $y=Tanx$ is defined as$x=Ta{n}^{-1}y$ or $x=Arc\tan y$. The domain of the arctangent function will be all real numbers, and its range is the set of angle measures between $-\frac{\mathrm{\pi }}{2}<y<\frac{\mathrm{\pi }}{2}$. The arctangent graph looks like this:

Arctangent graph, Marilú García De Taylor - StudySmarter Originals

If we graph all the inverse functions together, they look like this:

Arcsine, Arccosine, and Arctangent graphs together, Marilú García De Taylor - StudySmarter Originals

Please refer to the Inverse Trigonometric Functions article to learn more about this topic.