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

**values will be the trig value (like sin, cos, and tan).**

*y*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

**radians. The reason**

*π***tan**(and

**cot)**repeat after only

**radians is because, when dividing**

*π***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**** π** radians, and the

**period**for the

**tan**and

**cot**graphs is

**radians.**

*π*# 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**.

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.

# 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 \(\left[ -1,\,\,1 \right]\).

Note again that the graph starts repeating itself (cycles) every **2 π** radians (the

**); this is called the**

*x***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

**(**, where

*πk*, 0)*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 \(\left[ -1,\,\,1 \right]\).

Note again that the graph starts repeating itself (cycles) every **2**** π** radians (the

**); this is called the**

*x***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 \(x\ne \frac{\pi }{2}+\pi k\), *k *is an integer (where the asymptotes are), and the **range** is \(\left( -\infty ,\infty \right)\).

Note the **tangent** graph repeats itself (cycles) every ** π** radians (the

**); this is called the**

*x***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

**(**

*π*

*k***, 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 \(\displaystyle x\ne \pi k\), *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).

Note again that the graph starts repeating itself (cycles) every **2**** π** radians (the

**); this is called the**

*x***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

**, we have to divide by 0 to get the cosecant; this creates the asymptote. Vertical asymptotes were discussed here in the**

*π***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 \(x\ne \frac{\pi }{2}+\pi k\), *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).

**2**** π** radians (the

**); this is called the**

*x***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 \(x\ne \pi k\), *k *is an integer (where the asymptotes are), and the **range** is \(\left( -\infty ,\infty \right)\).

Note the **cotangent** graph repeats itself (cycles) every ** π** radians (the

**); this is called the**

*x***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!**

**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! **

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.

Thanks so much for writing, and please let me know how I can make the site better 😉 Lisa

What about something like a negative tan? for ex: f(x)= -tan(x)

What would that look like?

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

Good notes

Thank u for this

where is the table of this graph ?? in radian.

Thanks so much for pointing this out! I fixed it – is that what you wanted? http://www.shelovesmath.com/trigonometry/graphs-sine-cosine-tangent/#table

Lisa