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

Lesson 18: Addition of angles and multiple angle formulas

Topic: This lesson covers Chapter 18: Addition of angles and multiple angle formulas.

WeBWorK: There are two WeBWorK assignments on today’s material:Trigonometry – Sum and Difference Formulas

Trigonometry – Double and Half Angle Formulas

#### Lesson Notes (Notability – pdf):

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

**Question of the Day: **Suppose we know the values of the trig functions of two angles and . Can we use them to find the values of the trig functions of the angle ?

Today, we will see how we can do exactly that – the idea is to work with formulas that allow us to calculate, for example, and based on the values of and .

NOTE: We often use greek letters for angles — this helps us keep track of what’s an angle and what’s not. The most common are the greek letters alpha and beta . We’ll be using these instead of and .

Warning: the videos for today’s lecture are *quite long* – however, they consist almost entirely of examples, with a lot of explanation. Feel free to skip around, or to try the WeBWorK first (if you get stuck, the videos might help).

## Addition and Subtraction of Angles

**Proposition 18.1. **For any angles and ,

### Where did these formulas come from!?

*Great question! To answer it, you need to see the *proof* of these formulas – this appears in your book in Chapter 18.*

Now, we are going to see how these formulas let us calculate the values of trig functions at many different angles, based on just a few common angles (such as those listed in the table below – if you don’t know them, this is a great time to learn them!).

**Example 18.2.** Find the exact values of the trigonometric functions:

a)

b)

c)

*VIDEO: Example 18.2 applying angle sum and difference formulas*

## Double and Half Angles

**Proposition 18.5. **Let be any angle. Then we have the half-angle formulas:

and the double-angle formulas:

**Example 18.6.** Find the exact values of the trigonometric functions:

a)

b)

c)

*VIDEO: Example 18.6 applying half-angle formulas*

Example 18.7. Find the trigonometric functions of when has the properties below.

a) and is in quadrant 1

b) and is in quadrant 2

*VIDEO: Example 18.7 applying double-angle formulas*

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

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