# Trig Tables and Right Triangles

I realize that trig tables are somewhat out-of-fashion, but I think my question still makes sense in the computer age: why do trig tables treat the case of a right triangle instead of an arbitrary triangle?

Choose a triangle with altitude 1. Label the base angles a and b.

Sin a = 1/y   Sin b = 1/z
Tan a = 1/w   Tan b = 1/x


Solve each equation for the variable in the denominator:

y = 1/Sin a   z = 1/Sin b
w = 1/Tan a   x = 1/Tan b


This analysis leads to the creation of a trig table for general triangles. Where two angles are of measure a and b, the three sides will be in the proportion y : z : (w+x). If we divide these three parts by (w+x), the result is a proportion of two parts (usually) not equal to 1, and a third part always equal to 1 and therefore needless to specify.

I think that this statement of the proportion is easier to understand than the usual version. Is the table accurate? Does it make any difference whether we consider an acute, right or obtuse triangle? When have trig tables been done like this before, and why is this version not familiar? If the right triangle format is preferred because it is smaller and fits better on one page, perhaps that preference should be reconsidered in an era in which trig is almost always done on a calculator (or another computer)?

• You certainly could define a set of trig-like functions for non-right triangles but they would be need to be functions of 2 variables since having only one of the two angles $a$ and $b$ does not determine the triangle, even up to scaling. Also, these two variable functions would be expressible in terms of the usual single variable right triangle functions.