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 $a$ and $b$. Can we use them to find the values of the trig functions of the angle $a+b$?

Today, we will see how we can do exactly that – the idea is to work with formulas that allow us to calculate, for example, $\sin(a+b)$ and $\cos(a+b)$ based on the values of $\sin(a),\cos(a),\sin(b),$ and $\cos(b)$.

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 $\alpha$ and beta $\beta$. We’ll be using these instead of $a$ and $b$.

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 $\alpha$ and $\beta$,

- $\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$
- $\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta$
- $\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$
- $\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$
- $\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}$
- $\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}$

### 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) $\cos \left(\frac{\pi}{12}\right)$

b) $\tan \left(\frac{5 \pi}{12}\right)$

c) $\cos \left(\frac{11 \pi}{12}\right)$

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

## Double and Half Angles

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

$\sin \frac{\alpha}{2} =\pm \sqrt{\frac{1-\cos \alpha}{2}}$

$\cos \frac{\alpha}{2} =\pm \sqrt{\frac{1+\cos \alpha}{2}}$

$\tan \frac{\alpha}{2} =\frac{1-\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1+\cos \alpha}=\pm \sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}}$

and the double-angle formulas:

$\sin (2 \alpha) =2 \sin \alpha \cos \alpha$

$\cos (2 \alpha) =\cos ^{2} \alpha-\sin ^{2} \alpha=1-2 \sin ^{2} \alpha=2 \cos ^{2} \alpha-1$

$\tan (2 \alpha) &=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha}$

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

a) $\sin \left(\frac{\pi}{8}\right)$

b) $\cos \left(\frac{9 \pi}{8}\right)$

c) $\tan \left(\frac{\pi}{24}\right)$

*VIDEO: Example 18.6 applying half-angle formulas*

Example 18.7. Find the trigonometric functions of $2\alpha$ when $\alpha$ has the properties below.

a) $\sin (\alpha)=\frac{3}{5},$ and $\alpha$ is in quadrant 1

b) $\tan (\alpha)=\frac{12}{5},$ and $\alpha$ is in quadrant 2

*VIDEO: Example 18.7 applying double-angle formulas*

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