From Förberedande kurs i matematik 1

We start by drawing three auxiliary triangles, and calling the three vertical sides \displaystyle x,\ y and \displaystyle z, as shown in the figure.

Using the definition of cosine, we can work out \displaystyle x\text{ } and \displaystyle y from

\displaystyle x=a\cos \alpha

\displaystyle y=b\cos \beta

and, for the same reason, we know that \displaystyle z\text{ } satisfies the relation

\displaystyle z=l\cos \gamma

In addition, we know that the lengths \displaystyle x,\ y and \displaystyle z satisfy the equality

\displaystyle z=x-y

If we substitute in the expressions for \displaystyle x,\ y and \displaystyle z, we obtain the trigonometric equation

\displaystyle l\cos \gamma =a\cos \alpha -b\cos \beta

where \displaystyle \gamma is the only unknown.