From Förberedande kurs i matematik 1

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

Using the definition of cosine, we can work out x and y from

\displaystyle \begin{align} x &= a\cos \alpha\,,\\[3pt] y &= b\cos \beta\,, \end{align}

and, for the same reason, we know that z satisfies the relation

\displaystyle z=\ell\cos \gamma\,\textrm{.}

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

\displaystyle z=x-y\,\textrm{.}

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

\displaystyle \ell\cos \gamma = a\cos \alpha -b\cos \beta\,\textrm{,}

where \displaystyle \gamma is the only unknown.