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!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

**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