Sum-to-product sine calculator

Enter angles A and B in degrees or radians, and it applies sin(A) + sin(B) = 2 sin((A + B)/2) cos((A - B)/2). The result shows the substituted original expression, the factored expression, both half-angles, and independently evaluated values for the two equivalent forms. This makes it useful for simplifying trigonometric work, checking identity exercises, preparing equations, and exposing the average-angle and difference-angle structure hidden inside a sine sum.

Enter two angles in one shared unit

Begin with a sum containing exactly two sine terms, such as sin(50°) + sin(10°). Enter the first angle as A and the second as B, then choose degrees or radians as their shared unit. Both angles must be finite numbers because the calculator performs a numeric substitution and value check; it does not parse variables, fractions containing pi, or unrestricted algebraic expressions. Negative angles and angles beyond a complete revolution are accepted, since sine and cosine remain defined and periodic for those inputs. Do not mix units within one request. If A is measured in degrees, B must also be measured in degrees; if both values are radians, select rad. Any coefficient multiplying the entire original sum should remain outside the transformation and multiply the returned factored expression afterward. If the sine terms have separate coefficients, this particular identity does not apply directly unless earlier algebra makes those coefficients equal. These boundaries keep the transformation predictable and make every reported intermediate angle easy to audit.

Follow the average and half-difference

The calculator uses sin(A) + sin(B) = 2 sin((A + B)/2) cos((A - B)/2). It first adds the angles and divides by two, producing the half-sum or average angle used by the sine factor. It then subtracts B from A and divides by two, producing the half-difference used by the cosine factor. Finally, it multiplies those two trigonometric factors by 2. The returned half_sum_angle and half_difference_angle fields expose those intermediate results rather than hiding them inside a formatted answer. This is helpful because the most common manual errors are forgetting to divide both combined angles by two, reversing a sign, or omitting the leading factor of 2. Swapping A and B changes the sign of the half-difference, but it does not change the final product because cosine is an even function. Parentheses still matter: the division by two applies to the complete sum and the complete difference, not just to one input angle.

Check the value and use the factored form

The original_value field evaluates the two sine terms separately and adds them. The factored_value field evaluates the returned product independently. Those numbers should match apart from harmless floating-point effects, because the sum-to-product equation is an identity for every finite real pair of angles. Results are rounded to twelve decimal places, and values extremely close to zero are normalized to zero so familiar exact results do not appear as distracting machine artifacts. The product form can make trigonometric equations easier to solve because a product is zero when one of its factors is zero. It is also valuable when comparing waves, analyzing interference, deriving beat behavior, or simplifying an expression before another algebraic step. Keep the selected unit attached to both returned half-angles when carrying them into later calculations. For exact symbolic work involving pi or variables, use the identity field and the displayed factor structure as a template; the numeric fields are intended as a verification of the supplied finite angles.