Proving the Angle Addition Formulas

Lesson 1 of 2: Proof and Derivation Chain

In this lesson:

• Prove cos(A − B) using the unit circle and distance formula
• Derive cos(A + B), sin(A + B), and sin(A − B) by substitution
• See how one proof generates all four formulas

What You Will Learn Today

By the end of this lesson, you can:

1. State all four angle addition and subtraction formulas
2. Follow the geometric proof of using the distance formula
3. Derive , , and from

Hook: Does Sine Distribute over Addition?

Claim: ?

Test with , :

• Left:
• Right:

— the claim is false.

Setting Up the Proof on the Unit Circle

Points: ,

Step 1: Distance Formula for Chord PQ

Expand:

Step 2: Apply the Pythagorean Identity

From Step 1:

Since :

Step 3: Chord in Standard Position

The same chord placed at angle :

, compared to :

Step 4: Equate and Conclude

Both expressions equal :

Subtract 2 from both sides, then divide by :

Check: Verify the Formula Numerically

Use ,

Direct:

Formula:

Evaluate both terms and confirm equality.

Check Answer: cos Subtraction Formula Confirmed

The formula is confirmed.

Deriving cos(A + B): Replace B with −B

Apply with . Use even/odd properties:

Warning: The Sign for cos(A + B)

The sign in the formula is opposite to the sign in the argument.

Deriving sin(A + B): The Co-Function Bridge

Use the identity:

Apply with :

Completing the sin(A + B) Derivation

Substitute :

Check: Derive sin(A − B) in One Step

— apply with .

Using and :

Write the formula before advancing.

Check Answer: sin(A − B) Is Derived

One substitution — one new formula.

All Four Formulas: Reference Summary

Derived from one proof — not four separate facts.

Key Ideas from This Lesson

1. : proved from unit circle and distance formula
2. Three formulas: derived by substitution
3. Sign rule: addition has minus; subtraction has plus

(minus sign)

Coming Up in Lesson 2

Using the addition formulas:

• Decompose non-standard angles: 75° = 45° + 30°
• Find exact values of sin(75°), cos(15°), tan(105°)
• Derive double angle formulas from the addition formulas
• Apply: if , find