Enter two angles in radians — automatically convert product into sum/difference form
sin A × cos B
rad ×rad
cos A × sin B
rad ×rad
cos A × cos B
rad ×rad
sin A × sin B
rad ×rad
Results
Step-by-Step Derivation
Product-to-Sum Formulas
① sin A · cos B = (1/2)[sin(A+B) + sin(A-B)]
② cos A · sin B = (1/2)[sin(A+B) − sin(A-B)]
③ cos A · cos B = (1/2)[cos(A+B) + cos(A-B)]
④ sin A · sin B = (1/2)[cos(A-B) − cos(A+B)]
Product-to-sum formulas convert products of trigonometric functions into sums or differences, making integration, simplification, and equation solving much easier.
⚠Note: Input angles are in radians. 1 rad ≈ 57.3°, π rad = 180°, π/2 rad = 90°.
What Is Product-to-Sum?
Product-to-sum is an important trigonometric identity technique that converts the product of two trigonometric functions into a sum or difference. It is the inverse of sum-to-product, and together they simplify complex trigonometric expressions.
Derivation Method
From sin(A+B) = sinA cosB + cosA sinB and sin(A-B) = sinA cosB - cosA sinB. Adding gives 2 sinA cosB = sin(A+B) + sin(A-B).
Typical Application
Integral ∫ sin(ax) cos(bx) dx: use sin(ax) cos(bx) = (1/2)[sin((a+b)x) + sin((a-b)x)] to convert into two simple sine integrals.
Memory Trick
All have factor 1/2. sin·cos and cos·sin yield sine sum/difference; cos·cos yields cosine sum; sin·sin yields cosine difference (note the minus sign).
Inverse of Sum-to-Product
Sum-to-product: sin A + sin B → product form; Product-to-sum: product → sin A + sin B. They are inverse transformations.
① sin A · cos B = (1/2)[sin(A+B) + sin(A-B)]; ② cos A · sin B = (1/2)[sin(A+B) - sin(A-B)]; ③ cos A · cos B = (1/2)[cos(A+B) + cos(A-B)]; ④ sin A · sin B = (1/2)[cos(A-B) - cos(A+B)].
What is the relationship between product-to-sum and sum-to-product?▼
Product-to-sum and sum-to-product are inverse processes. Product-to-sum converts products like sin·cos, cos·cos into sum/difference form; sum-to-product converts sums like sin±sin, cos±cos into product form. They are mutually inverse transformations.
When are product-to-sum formulas most useful?▼
Product-to-sum formulas are essential for integration (e.g., ∫ sin(ax) cos(bx) dx becomes easier), signal processing (AM/FM modulation), wave superposition problems, and Fourier series expansions.
Why does sin A sin B have a minus sign?▼
From the identity cos(A-B) - cos(A+B) = -2 sin A sin B, we get sin A sin B = (cos(A-B) - cos(A+B))/2. The minus sign arises from the even symmetry of cosine (cos(-θ)=cosθ) and the derivation of sum/difference formulas.
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