SL 3.2—Triangle trigonometry

Choose the right triangle rule for a triangle problem.

Use Pythagoras for a right triangle; use sine/cosine rules or 1/2ab sin C when the triangle is not right-angled.

Worked example

With a=7,b=9 and C=60°, area is 1/2(7)(9)sin60°.

Name the known sides and included angle before choosing a formula.

The sine rule is not a universal replacement for the cosine rule.

Formula map: in a right triangle, $\sin A=opposite/hypotenuse$, $\cos A=adjacent/hypotenuse$, and $\tan A=opposite/adjacent$. For any triangle, $a/\sin A=b/\sin B=c/\sin C$, $c^2=a^2+b^2-2ab\cos C$, and $Area=\tfrac12ab\sin C$. With $a=7$, $b=9$, $C=60^\circ$, the cosine rule gives $c^2=49+81-126(1/2)=67$, hence $c=\sqrt{67}$, while the area is $63\sqrt3/4$ square units.