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 will be the angles (in radians – that’s the way it’s done), and the y values will be the trig value (like sin, cos, and tan).

The sin and cos (and csc and sec) functions start repeating after 2π radians, and it turns out that the tan and cot functions start repeating again after only π radians.    The reason tan (and cot) repeat after only π 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 2π radians, and the period for the tan and cot graphs is π radians.

Table of Trigonometric Parent Functions

Before we go into more detail of each of the trig functions, here is a table of the Trigonometric Parent Functions.   Note that when the domain can’t be certain values, we have asymptotes at those values of x.

Trig T-Charts

We will learn how to transform the trig functions in the Transformations of Trig Functions section.   Note that the t-charts I like to use for the Trig Functions can be found  here in the Transformations of Trig Functions section.

Trig Parent Functions

Graphs of the Six Trigonometric Functions

Here are the 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”.

 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  \(\left[ -1,\,\,1 \right]\).

Sine Function

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

Notice that the y intercept is (0, 0), and the x intercepts (zeros) are multiples of π; we can write this as (πk, 0), where k € I  (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:  \(\sin \left( -\frac{\pi }{2} \right)=-1=-\sin \left( \frac{\pi }{2} \right)\).

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  \(\left[ -1,\,\,1 \right]\).

Cosine Function

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

Notice that the y intercept is (0, 1), and the x intercepts are multiples of \(\frac{\pi }{2}\);  we can write this as \(\left( \frac{\pi }{2}+\pi k,\,\,0 \right)\), where k € I  (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:  \(\cos \left( -\frac{2\pi }{3} \right)=-\frac{1}{2}=\cos \left( \frac{2\pi }{3} \right).\)

Graph of Tan Function

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

Tangent Function

Note the tangent graph repeats itself (cycles) every π 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 π radians; this is because for multiples of  \(\frac{\pi }{2}\), 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 \(x=\frac{\pi }{2}+\pi k\); 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 π; we can write this as (πk, 0), where k € I  (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:  \(\tan \left( -\frac{\pi }{4} \right)=-1=-\tan \left( \frac{\pi }{4} \right)\).

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.

Here’s what a csc function graph looks like; notice that the domain is  \(\displaystyle x\ne \pi k\), 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 a y of 0, there are asymptotes for the csc function (since you can’t divide by 0).

Cosecant Function

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

Note also that there is a vertical asymptote every π radians; this is because for multiples of π, 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 \(\displaystyle x\ne \pi k\); thus the domain is all real numbers except for these asymptotes.

Notice that there is no y intercept, and no x intercepts; 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:  \(\csc \left( -\frac{\pi }{6} \right)=-2=-\csc \left( \frac{\pi }{6} \right)\).

Graph of Sec Function

Here’s what a sec function graph looks like; notice that the domain is  \(x\ne \frac{\pi }{2}+\pi k\), 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 a y of 0, there are asymptotes for the sec function (since you can’t divide by 0).

Secant Function

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

Note also that there is a vertical asymptote every π radians; this is because for multiples of  \(\frac{\pi }{2}\), 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  \(x=\frac{\pi }{2}+\pi k\); 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:  \(\sec \left( -\frac{\pi }{3} \right)=2=\sec \left( \frac{\pi }{3} \right)\).

Graph of Cot Function

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

Cotangent Function

Note the cotangent graph repeats itself (cycles) every π 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 radians; this is because for multiples of π, 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  \(\frac{\pi }{2}\); we can write this as  \(\left( \frac{\pi }{2}+\pi k,\,\,0 \right)\), where k € I  (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:  \(\cot \left( -\frac{\pi }{4} \right)=-1=-\cot \left( \frac{\pi }{4} \right)\).

Understand these problems, and practice, practice, practice!


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

8 thoughts on “Graphs of Trig Functions

  1. really I appreciated these notes well and I wish for my students and any other body to appreciate as well. Thank you how you prepared them.

    • You’d flip it across the x axis (actually since it’s an odd function, you can flip it across y access too). Does that make sense? Lisa

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