Mollweide's Equations

Definition

Mollweide’s Equations are trigonometric identities that relate all six parts of a triangle (three sides $a, b, c$ and three angles $A, B, C$). They are particularly useful for checking the accuracy of a solved triangle.

$\frac{a + b}{c} = \frac{\cos\left(\frac{A - B}{2}\right)}{\sin\left(\frac{C}{2}\right)}$

• How to read: “The ratio of a plus b to c is equal to the cosine of the quantity A minus B all divided by two, divided by the sine of C divided by two.”
• Meaning: Relates the sum of two sides to the third via half-angle trig functions of the opposite angles.

$\frac{a - b}{c} = \frac{\sin\left(\frac{A - B}{2}\right)}{\cos\left(\frac{C}{2}\right)}$

• How to read: “The ratio of a minus b to c is equal to the sine of the quantity A minus B all divided by two, divided by the cosine of C divided by two.”
• Meaning: The side difference over the third side links to the angle difference—complements the sum equation above.

Why It Matters

These equations provide a critical check for triangle calculations. In navigation and surveying, failing to verify results with Mollweide’s Equations can allow subtle errors in trigonometric calculations to propagate, leading to significant geographical misplacements.

Core Concepts

• Verification Tool: Unlike the Law of Sines or Cosines, which only use four parts of a triangle at a time, Mollweide’s Equations use all six. If the results of a triangle solution satisfy these equations, the solution is highly likely to be correct.
• Symmetry: The equations can be cyclicly permuted for any pair of sides ($b+c, c+a$, etc.).
• Relationship: They highlight the deep proportional link between the arithmetic sum/difference of sides and the half-angle trig ratios.

Connected Concepts

• Law of Sines
• Law of Cosines
• Product-to-Sum Formulas
• Sum-to-Product Formulas
• Triangle Classification