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Calculus III

Course: Trigonometry
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

Terms you should be able to 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.

Text Notes

While your textbook may introduce the graphs of the six trigonometric functions in pairs, I am introducing them all at one so that you can make connections between them from the start.

Mini-Lectures and Examples

Supplementary Resources (recommended)

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.

Watch Numberphile's Beautiful Trigonometry for a fun animation connecting trigonometric ratios to graphs.

Supplemental Resources (optional)

Read Intuitive Understanding of Sine Waves from Better Explained.

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

rev. 2021-04-05