Lesson 11.3

Sum and Difference Formulas

What is the exact value of ? It's not a special angle, but we can build it from ones that are.

Introduction

You know the exact values for . But what about ? The sum/difference formulas let us compute exact values by combining known angles.

Prerequisite Connection

You know exact values for and how to verify trig identities.

Today's Increment

We're learning formulas for , , and .

Why This Matters

These formulas are essential for deriving Double-Angle and Half-Angle formulas, and for solving trig equations in Calculus.

The Sum and Difference Formulas

Sine

Cosine

Tangent

Memory Tip: For cosine, the signs "flip" (plus inside → minus outside). For sine, they stay the same.

Worked Examples

Example 1: Finding an Exact Value

Find the exact value of .

Decompose the Angle

Apply Cosine Sum Formula

Substitute Known Values

Answer:

Example 2: Simplifying with Known Values

If (QI) and (QI), find .

Find Missing Trig Values

Use .

Apply Formula

Calculate

Answer:

Example 3: Proving an Identity (Advanced)

Prove:

Apply Cosine Difference Formula

Substitute Known Values

QED. This is the Cofunction Identity! ✓

Common Pitfalls

Thinking

This is FALSE. You must use the full formula with products of sines and cosines.

Mixing up the sign in the Cosine formula

Remember: Cosine Sum has a MINUS inside the product. Cosine Difference has a PLUS.

Real-World Application

Audio Engineering & Music Synthesis

Sound waves are sine waves. When two notes are played together, the resulting wave is described by sum formulas. This is the mathematical basis for understanding beat frequencies and harmonic mixing in music production software.