Solving a triangle with the law of cosines

When we know two sides and the included angle of a triangle, the law of cosines lets us immediately find the third side.

Example — Solving a triangle, side-angle-side

A triangle has sides $b=5$, $c=7$ and included angle $\widehat{A} = 60°$. Find the third side $a$.

$a^2 = 25 + 49 - 2\cdot 5\cdot 7\cdot\cos 60° = 74 - 70\cdot\tfrac12 = 74 - 35 = 39.$ So $a = \sqrt{39}\approx 6{,}24$.

A general triangle. By drawing the perpendicular $CH$ to the side $AB$, the triangle is split into two right-angled triangles.

In right-angled triangles the elementary trigonometric functions apply directly, and by combining them one recovers precisely the law of cosines.

Links

Topics: Triangle trigonometry
Concepts: Solving triangles · The law of cosines
Functions: Cosine
Methods: Solving triangles
Skills: Calculating · Synthetic geometry · Using formulae