2.4 · Right and oblique triangles: choosing a law

Match sides with their opposite angles

Use lowercase $\displaystyle a,b,c$ opposite angles $\displaystyle A,B,C$. In a right triangle, sine is opposite over hypotenuse and cosine is adjacent over hypotenuse. A diagram makes the reference angle explicit.

For nonright triangles, a known opposite side-angle pair suggests the Law of Sines. SSS or SAS data suggests the Law of Cosines.

Use the appropriate oblique-triangle law

The Law of Sines is $\displaystyle a/\sin A=b/\sin B=c/\sin C$. The Law of Cosines is $\displaystyle c^2=a^2+b^2-2ab\cos C$. The included angle in SAS belongs between the two known sides.

Triangle area can be found by $\displaystyle K=ab\sin C/2$ using two sides and their included angle. Keep enough precision in intermediate values if a decimal answer is required.

$$c^2=a^2+b^2-2ab\cos C$$

SSA may have two solutions

An inverse sine gives one angle, but its supplement may also fit. After finding $\displaystyle B$, check whether $\displaystyle 180^\circ-B$ is another candidate with a positive remaining angle.

If the computed sine exceeds $\displaystyle 1$, no triangle exists. If two candidates both satisfy the angle-sum condition, there are two triangles. Do not use SSA as a congruence rule.