Hi everyone! Read through the material below, watch the videos, and collect your questions.

Lesson 17: Trigonometric functions

Topic: This lesson covers Chapter 17: Trigonometric functions.

WeBWorK: There are five WeBWorK assignments on today’s material:`Trigonometry - Unit Circle`

,`Trigonometry - Graphing Amplitude`

,`Trigonometry - Graphing Period`

,`Trigonometry - Graphing Phase Shift`

, and`Trigonometry - Graphing Comprehensive`

Today we start trigonometric functions. We’ll begin with a review of the basics of trigonometry — if you remember everything about trigonometry, you can skip this part (but please don’t!). Then we’ll think about how these behave as functions, and look at their graphs.

#### Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

## Review of trigonometry

### a. Angles

We measure angles using two different systems, degrees and radians. In degrees, a full circle is . In radians, a full circle is . Thus, we have radians. If we consider a half-circle (divide both sides by two), we get:

Although most people are more familiar with degrees from their day-to-day lives, most mathematics at this level and above use radians. It will help you to get familiar with common angles in radians! The first video gives a review.

*VIDEO: Review of angles, radian measure*

### b. Trigonometric Functions

**Definition. **To define the trigonometric functions, we consider the following diagram: the initial side of an angle lies on the positive x-axis, and the terminal side of the angle passes through a point .

The distance from the origin to the point is (note: if we know and we can calculate by using the Pythagorean Theorem, ). Then the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) of are defined:

*VIDEO: Review – definitions of trigonometric functions*

Next, we need to know the values of the trig functions for some common angles. Once again, this is a review of material from previous courses – the following video will take you through some examples, but for more details I recommend checking out the videos on our video resource page.

*VIDEO: Values of trig functions at common angles*

This is the end of the ‘review’ part of the lesson. Need a little more help? Take a look at the videos on our video resource page.

## Graphs of trigonometric functions

Now we turn to the main idea of this lesson. We begin by looking at the graphs of the basic trig functions, , , and .

*VIDEO: Graphs of basic trig functions*

**Example 17.8.** Graph the following functions:

*VIDEO: Example 17.8 modifying the graphs of trig functions – amplitude, period, phase shift.*

**Definition 17.9**. Let be one of the functions:

The number is called the **amplitude**, the number is the **period**, and the number is called the **phase shift**.

That’s it for now. Take a look at the WeBWorK!

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