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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?

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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.

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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)?

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    $\begingroup$ 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. $\endgroup$ – Adam May 25 '17 at 19:21
  • $\begingroup$ @Adam As originally posted my trig table does have one angle along the left edge and a second angle along the right edge (as you subsequently suggested). I don't know what you mean when you say that "these two variable functions would be expressible in terms of the usual single variable right triangle functions", unless you refer to the derivation of the table (as first described in my question). Have you seen this table ever before? I haven't. Why is it never used? $\endgroup$ – Chaim May 26 '17 at 14:02
  • $\begingroup$ @Adam I made a careless misstatement. I have one variable along the left and the other not along the right (as I said), but along the top. And I'm too late to edit the first comment. $\endgroup$ – Chaim May 26 '17 at 14:16
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1) Are trig tables out of fashion, no. we just use trig tables that are hard wired into silicon these days.

2) the right angle triangle is by far the most important. whenever you split a vector into orthogonal components, for summation, you are using a right angle triangle. Splitting vectors into non-orthogonal components is generally far less useful.

3) use the sine or cosine rules for dealing with the few % of cases where someone actually cares about an arbitrary triangle. This was trivial even in the old days when slide rules were still in fashion, let alone with todays computers.

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