Transforming the double-angle formulas for cos gives the half-angle formulas, and combining the addition formulas gives the product-to-sum and sum-to-product formulas.
Half-Angle Formulas
- sin²(α/2) = (1 − cos α)/2
- cos²(α/2) = (1 + cos α)/2
- tan²(α/2) = (1 − cos α)/(1 + cos α)
Example ①
Find cos(π/8) using the half-angle formula.
Answer:cos²(π/8) = (1+cos(π/4))/2 = (1+√2/2)/2 = (2+√2)/4。
Since π/8 is acute, cos(π/8) = √(2+√2)/2
Find cos(π/8) using the half-angle formula.
Answer:cos²(π/8) = (1+cos(π/4))/2 = (1+√2/2)/2 = (2+√2)/4。
Since π/8 is acute, cos(π/8) = √(2+√2)/2
Product-to-Sum Formulas (Main Formulas)
- sin α cos β = [sin(α+β)+sin(α−β)]/2
- cos α cos β = [cos(α+β)+cos(α−β)]/2
- sin α sin β = −[cos(α+β)−cos(α−β)]/2
Sum-to-Product Formulas (Main Formulas)
- sin A + sin B = 2sin((A+B)/2)cos((A−B)/2)
- cos A + cos B = 2cos((A+B)/2)cos((A−B)/2)
Example ②
Calculate sin 75° + sin 15° using the sum-to-product formula.
Answer:2sin(45°)cos(30°) = 2 × (√2/2) × (√3/2) = √6/2
Calculate sin 75° + sin 15° using the sum-to-product formula.
Answer:2sin(45°)cos(30°) = 2 × (√2/2) × (√3/2) = √6/2
Key Point
- Be careful with signs in half-angle formulas: determine ± from the quadrant
- Product-to-sum and sum-to-product formulas can be derived from the addition formulas (understanding is more important than memorization)
Practice Problems
- Find sin(π/12) using the half-angle formula.
- Transform cos 3θ cos θ using a product-to-sum formula.
- Calculate sin 70° − sin 20° using a sum-to-product formula.
Answers and Explanations
- Answer:√(2−√3)/2
Explanation: sin²(π/12) = (1−cos(π/6))/2 = (1−√3/2)/2 = (2−√3)/4。Take the positive square root. - Answer:(cos 4θ + cos 2θ)/2
Explanation: cos α cos β = [cos(α+β)+cos(α−β)]/2。α=3θ, β=θ。 - Answer:2cos45°sin25° = √2 sin25°
Explanation: sin A−sin B = 2cos((A+B)/2)sin((A−B)/2)。A=70°, B=20°。