Lesson 2: Sum and Difference Formulas
Estimated time: 35-40 minutes
Learning Objectives
- State the sum and difference formulas for sin, cos, and tan
- Find exact values of non-standard angles (e.g., 15°, 75°)
- Simplify expressions using these formulas
- Verify identities using sum/difference formulas
The Formulas
Cosine:
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
Sine:
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
Tangent:
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
Memory tip for cosine: cos of a sum has a minus; cos of a difference has a plus (opposite of what you might expect!).
Finding Exact Values
Example 1: cos 75°
cos 75° = cos(45° + 30°) = cos 45 cos 30 − sin 45 sin 30
= (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4
Example 2: sin 15°
sin 15° = sin(45° − 30°) = sin 45 cos 30 − cos 45 sin 30
= (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4
Example 3: tan(A+B) Given Values
If tan A = 3/4 and tan B = 1/2, find tan(A+B).
tan(A+B) = (3/4 + 1/2)/(1 − (3/4)(1/2)) = (5/4)/(1 − 3/8) = (5/4)/(5/8) = 2
Simplifying Expressions
Example 4: Simplify cos(π − θ)
cos(π − θ) = cos π cos θ + sin π sin θ = (−1)cos θ + (0)sin θ = −cos θ
Check Your Understanding
1. Find exact value of sin 105°.
2. Simplify sin(π/2 + θ).
3. Find cos(π/4 − π/6).
Key Takeaways
- Cosine sum: cos(A+B) has a minus sign; difference has a plus.
- Sine sum: follows the operation sign.
- Use 30+45=75, 45−30=15, 60+45=105, etc. to find exact values.
- These formulas are essential for deriving double-angle and half-angle formulas.