Multiple Angle Formulas for Cosine We’re going to build them using lots of algebra (6.3, 6.4) (1)

POD to set it up If θ 1 = π/4 and θ 2 = π/6, find θ 1 + θ 2 and θ 1 - θ 2.

POD to set it up If θ 1 = π/4 and θ 2 = π/6, find θ 1 + θ 2 and θ 1 - θ 2. θ 1 + θ 2 = 5π/12 θ 1 - θ 2 = π/12 How would you find the exact trig ratios for the new angles?

The Handout In the unit circle, you have a number of exact values for angles like π/4 and π/6, but not for angles like 5π/12. With multiple angle formulas, however, you can derive exact trig values from constituent angles. In sections 6.3 and 6.4, we look at trig formulas for multiple angles. In 6.3 we will use formulas for addition and subtraction of any sized angles; in 6.4, we will use those formulas to build double- and half- angle formulas.

The Good News Is that you don’t have to memorize these formulas. You do need to be able to recognize which one to use.

Subtraction Formula for Cosine Today we look specifically at formulas involving cosine. Tomorrow, we’ll look at sine and tangent. Since the rest of the cosine formulas can be derived from the Subtraction Formula for Cosine, we will look at the proof for that formula here. You can follow along on the handout.

Initial Diagram In the first diagram, look at angles u and v. Notice how angle (u-v) is the same measure as angle w. It is simply rotated into standard position on a unit circle.

Assign coordinates Using the definition for sine and cosine on a unit circle,

Assign coordinates Notice where the points A,R, P, and Q are located. Their coordinates are

Using the Distance Formula The length from A to R equals the distance from P to Q. Using the distance formula, Use algebra to simplify this. Don’t worry if it looks a little messy.

Using the Distance Formula The length from A to R equals the distance from P to Q. Using the distance formula, Smooth algebra yields

Substitution Since we have a unit circle, cos 2 θ + sin 2 θ = 1. And we can substitute

Finally Remember that w is the same angle as u-v. Replace with values for sine and cosine already established. Done.

Addition Formula for Cosine Use the Subtraction Formula and change signs.

Double-Angle Formulas for Cosine What happens when u = v? Don’t forget the Pythagorean Identity here…

Half-Angle Identity Use one of the Addition Formulas.

Half-Angle Formula Use the Half-Angle Identity to make them. Why would these be called “half-angle?”

Use an addition formula Find the exact value of cos 7π/12. Start by finding two angles which add to 7π/12, and whose trig ratios we know. I’ll show two different break downs, but they arrive at the same result.

Use an addition formula Find the exact value of cos 7π/12.

Use a half-angle formula Find an exact value for cos 112.5°. We can start by doubling 112.5° to 225°.

Use a half-angle formula Find an exact value for cos 112.5°. Why the negative sign in front of the radical sign?