Sum & Difference Formulas (Sine)
. The correct expansion uses a powerful formula that opens up exact values for angles you've never memorized.
Introduction
The unit circle gives you exact values at , and their multiples. But what about or ? The sum and difference formulas let you break these “non-standard” angles into sums or differences of angles you already know.
Past Knowledge
Unit circle values for standard angles and all foundational identities.
Today's Goal
Apply formulas to evaluate non-standard angles.
Future Success
Double-angle formulas (5.13) are derived directly from these sum formulas.
Key Concepts
The Sine Sum & Difference Formulas
Memory Pattern
For sine sum/difference: the trig functions alternate (sin·cos + cos·sin). The sign in the middle matches the sign in the argument: uses , and uses .
Common Angle Decompositions
| Target Angle | Decomposition |
|---|---|
Worked Examples
Evaluating sin 75°
Question: Find the exact value of .
Step 1: Decompose:
Step 2: Apply the sum formula:
Step 3: Substitute known values:
Final Answer:
Evaluating sin 15°
Question: Find the exact value of .
Step 1: Decompose:
Step 2: Apply the difference formula:
Final Answer:
Using Given Values
Question: If (QI) and (QI), find .
Step 1: Find missing values using Pythagorean identity: ,
Step 2: Apply the sum formula:
Final Answer:
Common Pitfalls
Distributing Sine Over Addition
. Sine is not a linear function. You must use the sum formula. This is the single most common error in this unit.
Real-Life Applications
Music Production — Sound Interference
When two sound waves with different frequencies combine, the resulting waveform is described using the sine sum formula. Audio engineers use this to predict constructive interference (louder sound) and destructive interference (silence), which is the principle behind noise-canceling headphones.
Practice Quiz
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