Table of Trigonometric Parent Functions | Trig Functions in the Graphing Calculator |

Graphs of the Six Trigonometric Functions | More Practice |

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$” (**csc**, **cot**) have the ea**SY** asymptotes ($ 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$” (**csc**, **cot**) have the ea**SY** asymptotes ($ 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! **