The cosine rule | Mathematico

The cosine rule allows us to find missing sides or missing angles in any type of triangle.

Standard triangle

The cosine rule states that

a^{2} = b^{2} + c^{2} - 2 b c \cos A

The cosine rule starts like Pythagoras with $a^{2} = b^{2} + c^{2},$ but because this isn't generally a right-angled triangle we have the extra term $- 2 b c \cos A$ .

This would work with any of the sides at the front of the formula, so

\begin{aligned} b^{2} & = a^{2} + c^{2} - 2 a c \cos B \\ c^{2} & = a^{2} + b^{2} - 2 a b \cos C \end{aligned}

also work.

One of the problems will invite you to try and prove the cosine rule, but for now let's get used to using it.

A triangle has sides of length $3$ and $4,$ and the angle between these sides is $120^{\circ} .$ Find the length of the third side.
A triangle has sides of length $7$ and $k,$ and the angle between them is $110^{\circ} .$ Given that the third side of the triangle has length $12,$ find the value of $k .$
A triangle has sides of length $4, 8, 9.$ Find the largest angle in the triangle.