Non-Right Triangles — Sine Rule, Cosine Rule, and Area

Most real triangles are not right-angled. Land surveyors triangulating a plot, engineers analyzing a truss, and astronomers measuring stellar distances all work with oblique triangles. Three tools handle every such case: the sine rule, the cosine rule, and the area formula.

The sine rule

For any triangle with sides $a, b, c$ opposite angles $A, B, C$: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$$ Use the sine rule when you know two angles and any side (AAS/ASA) or two sides and a non-included angle (SSA). Write it with the unknown on top when solving for a side, and flipped ($\frac{\sin A}{a} = \dots$) when solving for an angle.

Worked example — triangulation. Two survey stations $P$ and $Q$ are $500$ m apart. From $P$, a landmark $R$ has an angle of $65^\circ$ measured from