You know, this ain’t the sort of thing Cecil would deal with. But I’ve looked at my old math textbooks and scoured the web and I can’t find a direct answer to what I seek.
I know the answer is pretty simple, but it’s so simple it’s not in my Calculus textbook, and it’s been so long since geometry that I don’t remember it.
Given the lengths of the sides of a right triangle determine the measurements of the other two angles. I think there’s a formula for this, but it’s slipped my mind.
I said it was simple didn’t I?
“SOH” “CAH” “TOA”
Sin (angle) = opposite side to angle / hypotenuse
Cosine (angle) = adjacent side to angle / hypotenuse
Tangent (angle) = opposite side to angle / adjacent side to angle
Exactly, but given the sin of the angle how do you determine the angle? There’s an inverse sin button on my calculator that works great, but I’m not sure what it’s doing.
Let’s say you get a sine of something about .707. If you take the inverse sine of that you should get approximately 45 degrees.
I know, but what the heck is an inverse sin? What is it doing?
The inverse sine function on your calculator is similar to opening up the back of a trigonometry textbook and letting you look up what the sine of an angle is. Instead of saying “Oh, the sine of a 45 degree angle is .707”, you are saying “if the sine is .707, the angle must be 45 degrees.”
Pulling this out of my very long ago memory, the value of a sine is always between 0 and 1.
Try putting 2 into your calculator and get the inverse of the sine. You should get an error.
Before the days of calculators average people used published tables for that sort of info, which gave you an approximation(well actually calculators give you an aproximation also, but usually to many more digits) from a zero starting point you do it iteratively.
Also, make sure you are not mixing radians and degrees. Sorry if that is obvious, but that sometimes gets me if someone else has been using my calculator, and has switched from radians to degrees without my knowledge.
So there isn’t simple formula to figure it out? Are you folks saying that there’s only some kind of iterative process that normal folks would never want to bother with like calculating pi?
That’s pretty much it.
Here’s what Wolfram and Mathematica have to say about it. Notice the formula labeled as “Maclaurin series.” Of course, they’re using radian measure, but that’s easy to convert to degrees.
Thanks everybody.
No wonder I couldn’t remember that formula! No one ever taught it to me. I guess it wasn’t that simple after all.
I seem to have a knack for poorly naming my threads…
It should be in your Calculus book. Look for a section on Taylor and Maclaurin series. (or serieses, or seri) If it doesn’t use the inverse sine function as an example, it will, at the very least, equip you with the tools to derive the series on your own. It is not as hard as it looks.
Don’t worry about how it works . . . but here is how it works, IIRC.
Let’s take Sine, which is the opposite leg divided by the hypotenuse. From Sine you can derive the ratio of the length of the Op leg (henceforth called O) to the Hypotenuse (henceforth called H). If you know the measurement of either leg you’ve got both. So now you have the angle and two of the sides. From the two sides you have the third (pythagorean theorum) and from the sides you have the angles.
On a somewhat unrelated note, I once wrote a calculator program to derive all the possible sides and angles for my trig class. Took 12 hours including debugging. Ah, those days were fun . . . watching the whole class do it by brute force while I punched in the numbers.
And BTW, what a problem about the angles and sides of a triangle would be doing in your calc book is something I can’t explain. Perhaps it would be weaving around the Y or X-axis to form a 3-dimensional object and you’d be asked to derive the speed at which it takes up space or something at T-4.
I unvolunteer myself for that problem.
If you want to do it by hand, then it’s a lot of work to get an approximation, by taking a bunch of terms of a series. On the other hand, your calculator can do most of the work for you. You specified that you have the legs, so we want the inverse tan function (usually called atan, arctan, or tan[sup]-1[/sup]). Let’s say that the legs you have are a and b, and the angles opposite those legs are A and B, respectively (note capitals). The formulas are then A = atan(a/b) and B = atan(b/a). Simple.
A word of warning on the inverse trig functions: The inverse trig functions are all multi-valued (not really functions), so your calculator has to guess which answer you want, and gives you what’s known as the “principle value” of the function. Always do a reality check when you get your answer, to make sure that that’s the one you want. For instance: The inverse sine of .707 is 45[sup]o[/sup]… or it could be 135[sup]o[/sup]. Many programming languages have a special version of atan designed to avoid this problem, by taking two arguments. It’s usually called atan2, and it takes the form theta=atan2(y,x) , where theta will be the angle counterclockwise from the positive x axis to the point (x,y).