Non-Right Triangles

Not every triangle contains a right angle, so we need tools that work for any triangle. The two central results are the sine rule and the cosine rule, supported by a compact area formula. Together they let you solve a triangle—that is, find all its unknown sides and angles—given enough information.

The sine rule. For a triangle with angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively,

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.$$

Use it when you know either two angles and any side (AAS or ASA) or two sides and a non-included angle (SSA). To find a side, put the sides on top: $a=\frac{b\sin A}{\sin B}$. To find an angle, invert the ratios: $\sin A=\frac{a\sin B}{b}$.

Worked example. In triangle $ABC$, $A=40^\circ$, $B=75^\circ$, and $a=10$. Then $C=65^\circ$, and $c=\fr