Angle from three sides

Problem

In a triangle $a=7$, $b=8$, $c=9$. Work out the angle $\widehat{A}$.

Solution

Isolate $\cos\widehat{A}$ from the cosine rule $a^2 = b^2 + c^2 - 2bc\cos\widehat{A}$: $\cos\widehat{A} = \frac{b^2 + c^2 - a^2}{2bc} = \frac{64 + 81 - 49}{2\cdot 8\cdot 9} = \frac{96}{144} = \frac{2}{3}.$ Hence $\widehat{A} = \arccos\tfrac{2}{3} \approx \boxed{48{,}19°}$.

Links

Topics: Triangle trigonometry
Concepts: Cosine rule
Functions: Cosine
Methods: Triangle solving
Skills: Calculating · Using formulae
Exercise type: Geometric problem