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Sum-to-Product Formulas Calculator

Online sum-to-product calculator: enter α, β to express sinα+sinβ as a product, the reverse of product-to-sum.

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sinα+sinβ=2sin((α+β)/2)cos((α−β)/2)
In degrees
In degrees

📖 Tutorial | Sum-to-Product

1. Definition
Express the sum (difference) of sines as a product: sinα+sinβ=2sin((α+β)/2)cos((α−β)/2).
2. Formula & Notation
α, βTwo angles
(α+β)/2Average angle
3. Properties & Laws
  • sinα+sinβ=2sin(avg)cos(half-diff);
  • sinα−sinβ=2cos(avg)sin(half-diff);
  • cosα+cosβ=2cos(avg)cos(half-diff);
  • cosα−cosβ=−2sin(avg)sin(half-diff).
4. Steps
  1. Enter α, β;
  2. Click "Calculate";
  3. Or click "Load Example".
5. Worked Examples
Example: sin75+sin15=2sin45cos30≈1.2247.
6. Common Pitfalls
Do not miss the factor 2;
One factor uses the average, one the half-difference;
cosα−cosβ has a minus.

❓ FAQ | Sum-to-Product

sinα+sinβ?
2sin(avg)cos(half-diff).
sinα−sinβ?
2cos(avg)sin(half-diff).
cosα+cosβ?
2cos(avg)cos(half-diff).
cosα−cosβ?
−2sin(avg)sin(half-diff).
Factor?
2.
Reverse of?
Product-to-sum.