Trigonometry Advanced

Product-to-Sum & Sum-to-Product Formulas

Convert between products and sums of trigonometric functions using the eight fundamental identities. Choose a direction, enter angles A and B, and see the symbolic formula, step-by-step substitution, and side-by-side numerical verification confirming the two sides are equal.

Product → Sum (4 formulas)
Sum → Product (4 formulas)
Degrees & Radians
Numerical Verification

Input

sin(A) · sin(B) = ½[cos(A−B) − cos(A+B)]
sin(A) + sin(B) = 2 sin((A+B)/2) cos((A−B)/2)

Result

Select a formula type, enter angles A and B, then click Convert.
Product-to-Sum Conversion
Numerical Verification
Product value
—
=
Sum value
—
Sum-to-Product Conversion
Numerical Verification
Sum value
—
=
Product value
—

Step-by-Step Solution

Formula Reference

All 8 product-to-sum and sum-to-product identities. Formulas in sky blue involve products; formulas in gold involve sums.

Product → Sum (4 formulas)
sin A · sin B
equals
½[cos(A−B) − cos(A+B)]
cos A · cos B
equals
½[cos(A−B) + cos(A+B)]
sin A · cos B
equals
½[sin(A+B) + sin(A−B)]
cos A · sin B
equals
½[sin(A+B) − sin(A−B)]
Sum → Product (4 formulas)
sin A + sin B
equals
2 sin((A+B)/2) cos((A−B)/2)
sin A − sin B
equals
2 cos((A+B)/2) sin((A−B)/2)
cos A + cos B
equals
2 cos((A+B)/2) cos((A−B)/2)
cos A − cos B
equals
−2 sin((A+B)/2) sin((A−B)/2)

How the formulas are derived

The product-to-sum identities come directly from adding or subtracting the sum and difference formulas for cosine and sine.

cos(A−B) = cosA cosB + sinA sinB
cos(A+B) = cosA cosB − sinA sinB

Add both equations: the sinA sinB terms cancel, giving 2 cosA cosB = cos(A−B) + cos(A+B), so cosA cosB = ½[cos(A−B) + cos(A+B)].

Subtract the second from the first: the cosA cosB terms cancel, giving 2 sinA sinB = cos(A−B) − cos(A+B). The sin formulas follow from the same trick applied to sin(A±B).

The sum-to-product formulas are just the same identities read in reverse after substituting u = (A+B)/2, v = (A−B)/2.

Applications

  • Evaluating integrals — products like sin x cos x are hard to integrate directly; converting to ½ sin 2x makes it immediate.
  • Solving trig equations — sums like sin 3x + sin x = 0 factor cleanly after converting to a product, revealing the zeros.
  • Signal processing — amplitude modulation (AM radio) multiplies a carrier wave by a signal wave; the product-to-sum identity reveals the two sideband frequencies.
  • Fourier analysis — orthogonality of sine/cosine is proved by showing that products integrate to zero using these identities.
  • Beat frequency — two near-frequency sound waves sum to a product of a slow beat envelope and a rapid oscillation, directly from sum-to-product.

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