Graphs of Trigonometric Functions

Note that limits of sine and cosine functions can be found here in the Limits and Continuity section.

Now that we know the Unit Circle inside out, let’s graph the trigonometric functions on the coordinate system. The $x$-values are the angles (in radians – that’s the way it’s done), and the $y$-values are the trig value (like sin, cos, and tan).
The sine and cosine (and cosecant and secant) functions start repeating after $2\pi$ radians, and the tangent and cotangent functions start repeating again after only $\pi$ radians. The reason tan (and cot) repeat after only $\pi$ radians is because, when dividing sin and cos to get tan, we get the same values in quadrants III as IV as we do for quadrants I and II, respectively. (Try this with the Unit Circle).

A complete repetition of the pattern of the function is called a cycle, and the period is the horizontal length of one complete cycle. Thus, the period of the sin, cos, csc, and sec graphs is $2\pi$ radians, and the period for the tan and cot graphs is $\pi$ radians. Because the trig functions are cyclical in nature, they are called periodic functions.

You may also hear the expressions sin wave and cos wave for the sin and cos graphs, since they look like “waves”.

Table of Trigonometric Parent Functions

The ftable below contains t-charts of the Trigonometric Parent Functions; this table is especially useful for the Transformations of Trig Functions section.

Note that when the domain can’t be certain values, there are vertical asymptotes at those values of $x$. (We learned about vertical asymptotes here in the Graphing Rational Functions, including Asymptotes). One way I remember the asymptotes: For the trig functions that have asymptotes, the functions that start with “$c$” (csc, cot) have the eaSY asymptotes ($x=\pi k$), while the other functions (tan, sec) have the more difficult ones ($\displaystyle x=\frac{\pi }{2}+\pi k$), where $k\in \mathbb{Z}$ ($k$ is the set of Integers).

Starting and stopping points may be changed, as long the graph covers one complete cycle (period).

Graphs of the Six Trigonometric Functions

Note that sin, csc, tan and cot functions are odd functions; we learned about Even and Odd Functions here. As an example, the sin graph is symmetrical about the origin $(0,0)$, meaning that if $(x,y)$ is a point on the function (graph), then so is $(-x,-y)$. It also means that for the sin graph, $f\left( -x \right)=-f\left( x \right)$. Let’s try this:  $\displaystyle \sin \left( -\frac{\pi }{2} \right)=-1=-\sin \left( \frac{\pi }{2} \right)$.

The cos and sec functions are even functions. As an example, the cos graph is symmetrical about the $y$-axis, meaning that if $(x,y)$ is a point on the function (graph), then so is $(-x,y)$. It also means that for the cos graph, $f\left( -x \right)=f\left( x \right)$. Let’s try this: $\displaystyle \cos \left( -\frac{2\pi }{3} \right)=-\frac{1}{2}=\cos \left( \frac{2\pi }{3} \right).$

For the csc function, notice the (dashed) sin function on the same graph; where the sin function has $y=0$, there are asymptotes for the csc function (since you can’t divide by 0). Similarly, for the sec function, the (dashed) cos function on the same graph; where the cos function has $y=0$, there are asymptotes for the sec function (since you can’t divide by 0).

Dotted lines represent the vertical asymptotes; remember again that the functions that start with “$c$” (csc, cot) have the eaSY asymptotes ($x=\pi k$), while the other functions (tan, sec) have the more difficult ones ($\displaystyle x=\frac{\pi }{2}+\pi k$).

Trig Functions in the Graphing Calculator

You can use the TI graphing calculator to graph trig functions, as follows:

There are also examples of using the calculator to solve trig equations here in the Solving Trigonometric Equations section.

Understand these problems, and practice, practice, practice!

For Practice: Use the Mathway widget below to try a Trig Graph problem. Click on Submit (the blue arrow to the right of the problem) to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Transformations of Trig Functions  – you’re ready!

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