The law of sines and law of cosines are the names of two formulas
that extend what we know about solving right triangles —
calculating side-lengths and angle measures —
to any triangle.
Among a triangle’s six features (three sides and three internal angles)
typically we need to know the measures of three of those features
to uniquely determine the entire triangle up to congruence.
Only “typically” though; there’s some nuance
depending on which three features we know.
For a triangle with internal angles measuring
\(\alpha\) and \(\beta\) and \(\gamma\)
that oppose sides of length \(A\) and \(B\) and \(C\) respectively,
we have the law of sines and the law of cosines:
\[
\frac{\sin(\alpha)}{A} = \frac{\sin(\beta)}{B} = \frac{\sin(\gamma)}{C}
\qquad \qquad C^2 = A^2 + B^2 - 2AB\cos(\gamma)
\]
The law of cosines quickly leads to another area formula. Letting \(S = \frac{1}{2}(A+B+C),\) a number referred to as the semiperimeter of the triangle, the area of the triangle will be \(\sqrt{S(S-A)(S-B)(S-C)}.\) This is called Heron’s formula.
Following quickly from the idea that cosine gives the percentage
by which a radius is scaled as is projects onto the polar (adjacent) axis,
we get these projection laws: