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 $360^\circ$. In radians, a full circle is $2\pi \text{rad}$. Thus, we have $360^\circ=2\pi$ radians. If we consider a half-circle (divide both sides by two), we get:

$$180^\circ=\pi$$

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 $x$ lies on the positive x-axis, and the terminal side of the angle passes through a point $P(a,b)$.

The distance from the origin $(0,0)$ to the point $P$ is $r$ (note: if we know $a$ and $bg$ we can calculate $r$ by using the Pythagorean Theorem, $r=\sqrt{a^2+b^2}$). Then the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) of $x$ are defined:

$$\sin(x)=\frac{b}{r},\quad \cos(x)=\frac{a}{r},\quad\tan(x)=\frac{a}{b}$$

$$\csc(x)=\frac{r}{b},\quad \sec(x)=\frac{r}{a},\quad\cot(x)=\frac{b}{a}$$

*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, $\sin x$, $\cos x$, and $\tan x$.

*VIDEO: Graphs of basic trig functions*

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

$f(x)=\sin (x)+3,\quad g(x)=4 \cdot \sin (x), \quad h(x)=\sin (x+2),\quad i(x)=\sin (3 x)$

$j(x)=2 \cdot \cos (x)+3, \quad k(x)=\cos (2 x-\pi), \quad l(x)=\tan (x+2)+3$

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

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

$f(x)=a \cdot \sin (b \cdot x+c) \quad \text { or } \quad f(x)=a \cdot \cos (b \cdot x+c)$

The number $|a|$ is called the **amplitude**, the number $\left|\frac{2 \pi}{b}\right|$ is the **period**, and the number $\frac{-c}{b}$ is called the **phase shift**.

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

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