Lesson 3: Double-Angle and Half-Angle Formulas
Estimated time: 35-40 minutes
Learning Objectives
- Derive double-angle formulas from sum formulas
- Apply three forms of cos 2θ
- Use half-angle formulas to find exact values
- Simplify expressions and verify identities using these formulas
Double-Angle Formulas
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan 2θ = 2 tan θ / (1 − tan²θ)
These follow directly from the sum formulas with A = B = θ.
Example 1: Finding sin 2θ
If sin θ = 3/5 and θ in QI, find sin 2θ.
cos θ = 4/5. sin 2θ = 2(3/5)(4/5) = 24/25.
Example 2: cos 2θ Three Ways
If sin θ = 3/5, find cos 2θ using all three forms.
cos²−sin² = 16/25 − 9/25 = 7/25. 2cos²−1 = 32/25−1 = 7/25. 1−2sin² = 1−18/25 = 7/25.
Power-Reducing Formulas
sin²θ = (1 − cos 2θ) / 2
cos²θ = (1 + cos 2θ) / 2
These are rearrangements of the cos 2θ formulas and are very useful in calculus.
Half-Angle Formulas
sin(θ/2) = ±√[(1 − cos θ) / 2]
cos(θ/2) = ±√[(1 + cos θ) / 2]
tan(θ/2) = (1 − cos θ) / sin θ = sin θ / (1 + cos θ)
Example 3: Exact Value Using Half-Angle
Find cos 15° using the half-angle formula (15 = 30/2).
cos 15° = √[(1 + cos 30)/2] = √[(1 + √3/2)/2] = √[(2+√3)/4] = √(2+√3)/2
Example 4: Half-Angle with Given Info
If cos θ = −3/5 and π < θ < 3π/2, find sin(θ/2).
θ/2 is in (π/2, 3π/4), so QII where sin is positive.
sin(θ/2) = +√[(1−(−3/5))/2] = √[(8/5)/2] = √(4/5) = 2/√5 = 2√5/5
Check Your Understanding
1. If cos θ = 12/13 (θ in QI), find sin 2θ.
2. Express cos²(3x) using a power-reducing formula.
3. Find tan 22.5° using a half-angle formula (22.5 = 45/2).
Key Takeaways
- sin 2θ = 2 sin θ cos θ is the most commonly used double-angle formula.
- cos 2θ has three equivalent forms — choose the one that simplifies your work.
- Half-angle formulas use ± based on the quadrant of θ/2.
- Power-reducing formulas lower the exponent by introducing 2θ.