Enter angles in radians to convert trigonometric sums/differences into product form
sin A + sin B
rad +rad
sin A − sin B
rad −rad
cos A + cos B
rad +rad
cos A − cos B
rad −rad
Results
Step-by-Step Derivation
Sum-to-Product Formulas
① sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)
② sin A − sin B = 2 cos((A+B)/2) sin((A-B)/2)
③ cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
④ cos A − cos B = −2 sin((A+B)/2) sin((A-B)/2)
Sum-to-product identities convert sums or differences of sine/cosine functions into products for easier calculation and integration.
⚠Note: Input values are in radians. 1 rad ≈ 57.3°, π rad = 180°, π/2 rad = 90°.
What Are Sum-to-Product Identities?
Sum-to-product is a key trigonometric identity technique that converts sums or differences of sine/cosine functions into product form to simplify calculations.
Memory Rule
Sum-to-product patterns: sine sum → sine-cosine product; sine difference → cosine-sine product; cosine sum → cosine product; cosine difference → negative sine product.
Derivation Principle
Derived from angle addition formulas: sin(A+B) = sinA cosB + cosA sinB and sin(A-B) = sinA cosB - cosA sinB. Add them to eliminate cross terms.
Example Usage
sin 30° + sin 60° = 2 sin 45° cos(-15°) = 2 × (√2/2) × cos 15° = √2 cos 15°.
Inverse to Product-to-Sum
Sum-to-product: sinA + sinB → product; Product-to-sum: sinA · cosB → sum. Used together to simplify complex trig expressions.
① sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2); ② sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2); ③ cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2); ④ cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2).
What is the relationship between sum-to-product and product-to-sum?▼
Sum-to-product and product-to-sum are inverse processes. Sum-to-product converts sin±sin, cos±cos into products; product-to-sum converts products like sin·cos, cos·cos back into sums or differences.
When to use sum-to-product identities?▼
Sum-to-product is widely used for integral simplification, solving trigonometric equations, Fourier transforms, and trigonometric series summation. It simplifies expressions by converting sums into single trigonometric products.
Why is there a coefficient of 2 in sum-to-product formulas?▼
The coefficient 2 comes from angle-addition identities. Expand sin A and sin B using sum/difference formulas, add them together, and factor out the common term to get 2 sin((A+B)/2) cos((A-B)/2).
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