Graphs of Trig Functions

This Graphs of Trig Functions section covers :

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 sin and cos (and csc and sec) functions start repeating after \(2\pi \) radians, and it turns out that the tan and cot 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. (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.

Table of Trigonometric Parent Functions

Before we go into more detail of each of the trig functions, here are some tables that might help. The first is a table of the t-charts of the Trigonometric Parent Functions; this table is especially useful for the Transformations of Trig Functions section. The second table shows more details for each of the trig functions.

Note that when the domain can’t be certain values, there are asymptotes at those values of \(x\). 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 (\(\displaystyle x=\frac{\pi }{2}+\pi k\)).

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

Trig Function T-Charts

\(y=\sin \left( x \right)\)

 

x

y

0  0
 \(\displaystyle \frac{\pi }{2}\)  1
\(\pi\)  0
 \(\displaystyle \frac{{3\pi }}{2}\) –1
\(2\pi\)  0

\(y=\cos \left( x \right)\)

 

x y
0   1
\(\displaystyle \frac{\pi }{2}\)   0
\(\pi\) –1
\(\displaystyle \frac{{3\pi }}{2}\)   0
\(2\pi\)   1
 

\(y=\tan \left( x \right)\)

 

x y
 \(\displaystyle -\frac{\pi }{2}\) undef
 \(\displaystyle -\frac{\pi }{4}\) –1
 0 0
\(\displaystyle \frac{\pi }{4}\) 1
\(\displaystyle \frac{\pi }{2}\) undef

\(y=\csc \left( x \right)\)

 

x y
0 undef
 \(\displaystyle \frac{\pi }{2}\) 1
\(\pi\) undef
 \(\displaystyle \frac{{3\pi }}{2}\) –1
\(2\pi\) undef

\(y=\sec \left( x \right)\)

 

x y
 0 1
 \(\displaystyle \frac{\pi }{2}\) undef
\(\pi \) –1
\(\displaystyle \frac{{3\pi }}{2}\) undef
\(2\pi \) 1

\(y=\cot \left( x \right)\)

 

x y
0 undef
 \(\displaystyle \frac{\pi }{4}\) 1
  \(\displaystyle  \frac{\pi }{2}\) 0
\(\displaystyle \frac{{3\pi }}{4}\) –1
\(\pi \) undef
Trig Function

\(k\in \text{ Integers}\)

Graph Trig Function

\(k\in \text{ Integers}\)

Graph

\(y=\sin \left( x \right)\)

Odd

Domain: \(\left( {-\infty ,\infty } \right)\)

Range: \(\displaystyle \left[ {-1,1} \right]\)

Period: \(2\pi \)

Zeros: \(\left( {\pi k,0} \right)\)

\(y=\csc \left( x \right)\)

Odd

 Domain: \(x\ne \pi k\)

Range: \(\displaystyle \left( {-\infty ,-1} \right]\cup \left[ {1,\infty } \right)\)

Asymptotes: \(x=\pi k\)

Period: \(2\pi \)

Zeros: None

\(y=\cos \left( x \right)\)

Even

Domain: \(\left( {-\infty ,\infty } \right)\)

Range: \(\displaystyle \left[ {-1,1} \right]\)

Period: \(2\pi \)

Zeros: \(\displaystyle \left( {\frac{\pi }{2}+\pi k,0} \right)\)

 

\(y=\sec \left( x \right)\)

Even

Domain: \(\displaystyle x\ne \frac{\pi }{2}+\pi k\)

Range: \(\displaystyle \left( {-\infty ,-1} \right]\cup \left[ {1,\infty } \right)\)

Asymptotes: \(\displaystyle x=\frac{\pi }{2}+\pi k\)

Period: \(2\pi \)

Zeros: None

\(y=\tan \left( x \right)\)

Odd

Domain: \(\displaystyle x\ne \frac{\pi }{2}+\pi k\)

Range: \(\left( {-\infty ,\infty } \right)\)

Asymptotes: \(\displaystyle x=\frac{\pi }{2}+\pi k\)

Period: \(\pi \)

Zeros: \(\left( {\pi k,0} \right)\)

\(y=\cot \left( x \right)\)

Odd

Domain: \(x\ne \pi k\)

Range: \(\left( {-\infty ,\infty } \right)\)

Asymptotes: \(x=\pi k\)

Period: \(\pi \)

Zeros: \(\displaystyle \left( {\frac{\pi }{2}+\pi k,0} \right)\)

 

Graphs of the Six Trigonometric Functions

Here are more detailed graphs of the six trig Functions. You may also hear the expressions sine wave and cosine wave for the sin and cos graphs, since they look like “waves”.

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 Graph of Sin Function

Here’s what a sin function graph looks like; notice that the domain is \(\left( -\infty ,\infty \right)\), and the range is \(\left[ -1,1 \right]\).

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Note again that the graph starts repeating itself (cycles) every \(2\pi \) radians (the \(x\)); this is called the period of the graph. This is one complete revolution around the unit circle.

The \(y\) intercept is \((0,0)\), and the \(x\)-intercepts (zeros) are multiples of \(2\pi\); we can write this as \(\left( {\pi k,0} \right)\), where \(k\in \mathbb{Z}\) (\(k\) is in the set of Integers).

The sin function is an odd function; we learned about Even and Odd Functions here. Thus, 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:  \(\displaystyle \sin \left( -\frac{\pi }{2} \right)=-1=-\sin \left( \frac{\pi }{2} \right)\).

Graph of Cos Function

Now let’s graph the cos function; notice again that the domain is \(\left( -\infty ,\infty \right)\), and the range is \(\left[ -1,1 \right]\).

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Note again that the graph starts repeating itself (cycles) every \(2\pi \) radians (the \(x\)); this is called the period of the graph. This is one complete revolution around the unit circle.

The \(y\)-intercept is \((0,1)\), and the \(x\)-intercepts are multiples of \(\displaystyle \frac{\pi }{2}\); we can write this as \(\displaystyle \left( \frac{\pi }{2}+\pi k,0 \right)\), where \(k\in \mathbb{Z}\) (\(k\) is in the set of Integers).

The cos function is an even function; we learned about Even and Odd Functions here. Thus, 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:  \(\displaystyle \cos \left( -\frac{2\pi }{3} \right)=-\frac{1}{2}=\cos \left( \frac{2\pi }{3} \right).\)

Graph of Tan Function

Here’s what a tan function graph looks like; notice that the domain is \(\displaystyle x\ne \frac{\pi }{2}+\pi k\), \(k\) is an integer (where the asymptotes are), and the range is \(\left( -\infty ,\infty \right)\).

Note the tangent graph repeats itself (cycles) every \(\pi\) radians (the \(x\)); this is called the period of the graph. This graph repeats itself one-half of a revolution of the unit circle.

Note also that there is a vertical asymptote every \(\pi\) radians; this is because for multiples of \(\displaystyle \frac{\pi }{2}\), we have to divide by 0 to get the tangent; this creates the asymptote. Vertical asymptotes were discussed here in the Graphing Rational Functions, including Asymptotes section. The vertical asymptotes on the tan graph are at \(\displaystyle x=\frac{\pi }{2}+\pi k\); thus, the domain is all real numbers except for these asymptotes.

There are no \(y\)-intercepts, and the \(x\)-intercepts are multiples of \(\pi\); we can write this as \(\left( {\pi k,0} \right)\), where \(k\in \mathbb{Z}\) (\(k\) is in the set of Integers).

The tan function is an odd function; we learned about Even and Odd Functions here. Thus, the tan 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 tan graph, \(f\left( -x \right)=-f\left( x \right)\). Let’s try this:  \(\displaystyle \tan \left( -\frac{\pi }{4} \right)=-1=-\tan \left( \frac{\pi }{4} \right)\).

Graph of Csc Function

The csc, sec, and cot are sometimes called the reciprocal functions, since they are the reciprocal of sin, cos, and tan, respectively. To draw them, it’s good to draw the original function first, and then overlay the reciprocal.

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Here’s what a csc function graph looks like; notice that the domain is \(\displaystyle x\ne \pi k\), \(k\) is an integer (where the asymptotes are), and the range is \(\displaystyle \left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)\). 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).

Note again that the graph starts repeating itself (cycles) every \(2\pi \) radians (the \(x\)); this is called the period of the graph. This is one complete revolution around the unit circle.

Note also that there is a vertical asymptote every \(2\pi\) radians; this is because for multiples of \(2\pi\), we have to divide by 0 to get the cosecant; this creates the asymptote. Vertical asymptotes were discussed here in the Graphing Rational Functions, including Asymptotes. The vertical asymptotes on the csc graph are at \(x=\pi k\); thus, the domain is all real numbers except for these asymptotes.

There are no \(y\)-intercepts and no \(x\)-intercept; the closest the graph gets to the \(x\)-axis is –1 and 1.

The csc function is an odd function; we learned about Even and Odd Functions here. Thus, the csc 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 csc graph, \(f\left( -x \right)=-f\left( x \right)\). Let’s try this:  \(\displaystyle \csc \left( -\frac{\pi }{6} \right)=-2=-\csc \left( \frac{\pi }{6} \right)\).

Graph of Sec Function

Here’s what a sec function graph looks like; notice that the domain is \(\displaystyle x\ne \frac{\pi }{2}+\pi k\), \(k\) is an integer (where the asymptotes are – same as tan function), and the range is \(\displaystyle \left( -\infty ,-1 \right]\cup \left[ 1,\infty \right)\) (same as the csc function).

Notice 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).

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Note again that the graph starts repeating itself (cycles) every \(2\pi \) radians (the \(x\)); this is called the period of the graph. This is one complete revolution around the unit circle.

Note also that there is a vertical asymptote every \(\pi\) radians; this is because for multiples of \(\displaystyle \frac{\pi }{2}\), we have to divide by 0 to get the secant; this creates the asymptote. Vertical asymptotes were discussed here in the Graphing Rational Functions, including Asymptotes section. The vertical asymptotes on the sec graph are at \(\displaystyle x=\frac{\pi }{2}+\pi k\); thus, the domain is all real numbers except for these asymptotes.

Notice that the \(y\)-intercept is \((0,1)\), and there are no \(x\)-intercepts; the closest the graph gets to the \(x\)-axis is  –1 and 1.

The sec function is an even function; we learned about Even and Odd Functions here. Thus, the sec 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 sec graph, \(f\left( -x \right)=f\left( x \right)\). Let’s try this:  \(\displaystyle \sec \left( -\frac{\pi }{3} \right)=2=\sec \left( \frac{\pi }{3} \right)\).

Graph of Cot Function

Here’s what a cot function graph looks like; notice that the domain is  \(x\ne \pi k\), \(k\) is an integer (where the asymptotes are), and the range is \(\left( -\infty ,\infty \right)\).

Note the cotangent graph repeats itself (cycles) every \(\pi\) radians (the \(x\)); this is called the period of the graph. The graph repeats itself one-half of a revolution of the unit circle.

Note also that there is a vertical asymptote every \(\pi\) radians; this is because for multiples of \(\pi\), we have to divide by 0 to get the tangent; this creates the asymptote. Vertical asymptotes were discussed here in the Graphing Rational Functions, including Asymptotes section. The vertical asymptotes on the cot graph are at \(x=\pi k\); thus, the domain is all real numbers except for these asymptotes.

Notice that there is no \(y\)-intercept, and the \(x\)-intercepts are multiples of \(\displaystyle \frac{\pi }{2}\); we can write this as \(\displaystyle \left( \frac{\pi }{2}+\pi k,0 \right)\), where \(k\in \mathbb{Z}\) ( \(k\) is in the set of Integers).

Note that the cot function is an odd function; we learned about Even and Odd Functions here. Thus, the cot 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 cot graph, \(f\left( -x \right)=-f\left( x \right)\). Let’s try this:  \(\displaystyle \cot \left( -\frac{\pi }{4} \right)=-1=-\cot \left( \frac{\pi }{4} \right)\).

Trig Functions in the Graphing Calculator

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

Graphing Trig Functions Instructions Screens
Push “Y=” and enter the trig equation in Y1 using the sin, cos, or tan buttons (followed by X,T,Ɵ,n). (You don’t need to close the parentheses after the \(x\), unless you’re doing more calculations).

 

To get one revolution, use the “window” and use 0 (Xmin) to \(2\pi \) (Xmax) (you should use radian mode). Note that for the Xmax, for example, you can actually use “2nd ^” to input \(\pi \).

 

You might want to use “ZOOM 6” (ZoomStandard) and “ZOOM 0” (ZoomFit) before working with the WINDOW, if the graph seems a little off-centered. You can also use “ZOOM 7” (ZTrig) to get a nice display of the graph.

 

We can check a complete revolution of the graph in a graphing calculator – looks good!

For the reciprocal functions csc, sec, and cot, use “1/sin(x)”, “1/cos(x)”, and “1/tan(x)”, respectively.

 

For the cotangent graph in the example on the right, I used “ZOOM 7” (Ztrig) to graph; it graphs it nicely.

 

 

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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