Find an Angle of a Right Triangle Without Trigonometric Functions

Question

I have a right triangle triangle. I know the length of the hypotenuse (H) and one adjacent side (A). I would like to find the angle between the A and the H without using $\arccos(A/H)$. I would like to avoid all trigonometric equations. Is there a theorem or method to find this angle without using trig?

Here is the general problem I am trying to solve:

I am unable to factor the acos() function to isolate H. Perhaps this is due to my limited experience with math. If so, is there a resource I should look into or another way to solve the problem

Answer 1

You can avoid using the trigonometric functions, by using tables of known values or some approximation of them (series expansions, solving certain algebraic equations for some cases etc.).

But in a sense you use them indirectly. It might help if you give some more background. Like you code some game in assembler and have no math library available.

Update:

We have the arc length $D = R \varphi$ and thus $$ D = R \arccos\left(\frac{R-H}{R}\right) $$ We can invert $D(H)$ to $H(D)$ by $$ H = R\left(1-\cos\left(\frac{D}{R}\right)\right) $$ Here is a plot for $x = D/R = 1 + \delta$ $H(x)$ and two approximations:

Answer 2

Except for special cases like the 30-60-90 and 45-45-90 triangles, finding angles from the sides in finitely many steps is a definition of trigonometric equations. So no, you can't do it in general.

If you allow infinite series or their approximations, you could do it--but basically you would be using calculus combined with trigonometry. Or you could use the trig ratios for a few angles and combine them. Those two methods are how computers and calculators do trigonometric ratios.