How to calculate angle if 2 sides are known?

[Fran3]

Given a right triangle with 2 known sides...

For instance you know...
-- the opposite = y
-- the adjacent =x
-- the angel = theta
-- theta >= 0
-- theta <= 90

Given the absence of a calculator or lookup table...

How would one go about calculating the angle theta from scratch?

Thanks for any help

 

[Dr.Peterson]

You would need to approximate the inverse tangent; you can do more or less what a calculator would do, or the people who made the tables. There are various ways to do that, such as series, or continued fractions, as well as algorithms used in calculators. See also here, starting particularly at equations 11 and 41.

But there is a good reason no one does this by hand! We let others (table makers or calculator designers) do the work for us.

 

[Fran3]

Thanks.
I was hoping for a more straight forward solution using simple algebra.
Especially since folks often study trig before they get to series & limits in calculus.
Actually I was surprised that I did not know/remember the answer to the question.
Things seem to get lost in the cobwebs of our mind it not taken out in the sun sometimes
Thanks again.

 

[Dr.Peterson]

There's a reason they used to give us trig tables, rather than a formula. Algebra isn't enough; people spent their lives making tables! Now we just spend a moment pushing a button.

But it would be nice if textbook authors at least mentioned to students something about what the calculator has to do, and why they can't do it by hand (not that many want to do so). (Some, I'm sure, do, in a sidebar no one reads.) I've heard from some students who evidently imagined that the calculator draws a triangle internally and takes some measurements ...