Mollweide's formula

Mollweide's formulas are two triangle identities relating sums and differences of side lengths to tangents of half-sums and half-differences of opposite angles.

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Sine rule: A triangle law stating that each side divided by the sine of its opposite angle has the same value. Replacing side lengths with opposite-angle sines gives a direct route to both identities.
Triangle: A polygon with three sides, three vertices, and three interior angles. Mollweide's formulas apply to the side-angle relationships of this basic figure.
Triangle solving: The process of determining a triangle's unknown sides and angles from given measurements. Mollweide's formulas can supply relations among known and unknown triangle measurements.
Carl Brandan Mollweide: A German mathematician and astronomer who lived from 1774 to 1825. The formulas bear his name and are associated with his mathematical work.
Angle sum of a triangle: The interior angles of a Euclidean triangle sum to 180 degrees, or π radians. This constraint converts sums and differences of angles into the half-angle expressions in the formulas.
Opposite angle: In a triangle, the angle across from a specified side. Each side in the identities is paired with its opposite angle.
Trigonometric identity: An equation involving trigonometric functions that holds for every value in its domain. Mollweide's formulas are triangle-specific identities within this broader class.
History of trigonometry: The historical development of methods for relating angles, lengths, and functions. Mollweide's identities belong to the long development of triangle trigonometry.
Tangent addition formula: An identity expressing the tangent of a sum or difference through the tangents of its component angles. It supplies an alternative algebraic route from angle sums to Mollweide's tangent ratios.
Semiperimeter: Half the sum of a polygon's side lengths; for a triangle, the sum of its three sides divided by two. The side sums in one identity can be rewritten using the triangle's semiperimeter.