Topic: Graphs of Trigonometric Functions

Subtopic: Basic Graphs

**Overview**

We have studied the trigonometric functions from an algebraic perspective, but today we begin to study and analyze their graphs. This chapter is graph intensive. You should have your electronic grapher ready so you can follow along. If you have any trouble producing the graphs on your grapher, please ask questions in class.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 4.1.1 Be able to graph the six basic trig functions (algebraically and electronically)
- 4.1.2 Know the domain, range, asymptotes, amplitude, period, and symmetries of the six basic trig functions
- 4.1.3 Recall from algebra that even functions satisfy f(-x)=f(x) and their graphs are symmetric about the y-axis, odd functions satisfy f(-x)=-f(x) and their graphs are symmetric about the origin. Use symmetry to determine graphically which of the six trig functions are even and which are odd.

**Terminology**

Define: domain, range, asymptote line, amplitude, period, symmetry, line of symmetry, symmetry (about x-axis, y-axis, origin, line y=x), odd function vs. even function.

**Supplementary Resources**

Download: Graphs of Six Trig Functions and Basic Information

Note the graphs are in blue with one period of each marked in bold blue. Asymptotes are in red. The last two functions show the sine and cosine graphs in green only for reference.

Check out the connection between the unit circle and the graph of the sine, cosine, and tangent functions at Interactive Unit Circle from Math Is Fun. As you drag the point around the unit circle, watch the length of the red "height" line defining the "height" of the sine function and the graph of a sine wave spin out. You can do similarly for the blue cosine function and green tangent function. Cool connections to understand, strongly recommended interaction!

If you prefer to see them separated check out Unit Circle and Sine Graph and Unit Circle and Cosine Graph each from GeoGebra.

A similar, but more complex, Java-based example is shown at Sine Wave Geometry from Dynamic Geometry.

For the other trig functions and their connection to the circle, see Graphs of Tangent, Cotangent, Secant, and Cosecant from Interactive Mathematics.

You may also want to read Intuitive Understanding of Sine Waves from Better Explained but it is completely optional.

**Supplemental Resources (optional)**

Videos from James Sousa's MathIsPower4U:

Mini-Lesson: __Graphing the Sine and Cosine Functions__

Animation: __Graphing the Sine Function Using the Unit Circle__

Animation: __Graphing the Cosine Function Using the Unit Circle__

Example: __Graph the Sine Function Using the Unit Circle__

Mini-Lesson: __Graphing Cosine, Sine, and Tangent on the TI84__

Mini-Lesson: __Graphing the Tangent Function__

Animation: __Graphing the Tangent Function Using the Unit Circle__

Mini-Lesson: __Graphing Tangent and Cotangent over different Periods__

Example: __Graphing the Tangent Function Using the Unit Circle and the Reciprocal Identity__

Example: __Graphing the Tangent Function Over a Different Period__

Mini-Lesson: __Graphing the Cosecant and Secant Functions__

Mini-Lesson:__Graphing the Cotangent Function__

Mini-Lesson:__Graphing Secant, Cosecant, and Cotangent on the TI84__

Example: __Graphing the Secant Function Using the Cosine Function__

Example: __Graphing the Cosecant Function Using the Sine Function__