## Thursday, September 08, 2011

### Relearning Math

I often forget that I don't really do hands-on mathematics anymore. By "hands-on," I mean doing it step-by-step; possibly by hand! A few years ago, I had a go at long division, but after several tries, gave up for a bit in frustration, had a coffee, mulled over it some, and then finished what should have been the operation of just a few minutes (if that).

Today, it was trigonometry. Specifically, it was trying to remember how to convert field measurements of elevations and point-to-point distances into point-to-point slopes. It wasn't too difficult to start off: get the difference of two consecutive elevations to obtain the change in height, and use the length between the two points as the -- you guessed it -- length. Okay, easy part done. Now... how does one apply "SOH, CAH, TOA" again?

Oh, right: SOH refers to the relationship in which the sine of the angle is equal to the length of the side of the triangle opposite the angle in question divided by the length of the hypotenuse of the triangle, or:

sine ϴ = opposite/hypotenuse

However, since I didn't have the value for ϴ, I needed to take the arcsine (or inverse sine) of the quotient, or:

ϴ = arcsine (opposite/hypotenuse)

Yeah, not too difficult. However, when I tried this out using Excel for my 30-60-90 triangle with lengths of 3-5.2-6, I divided 3 by 6, took the arcsine and got "0.523599" instead of the 30 that I was expecting. Ah, crap. What did I do wrong? Check it again by dividing the 5.2 by 6, take the arcsine and get "1.047198". Shoot. Well, at least the first was 1/2 that of the second, so the ratio is correct. However, the first is not 30 and the second isn't 60 degrees.

I knew that it was coming to this: hit F1 and subject myself to the Microsoft Office Help functions. ... where I was dutifully reminded that Excel does these calculations in radians and not degrees. Suddenly vague memories of trying to change my Casio graphing calculator from radians to degrees swam through my head. How does one do that transformation again? Go to Google.

Search "radians to degrees formula", and bypass all the calculators that don't actually have the formula (and yet somehow get placed in the search results). Ah, right:

Okay, so now, I go back to Excel, and multiply my two initial results by 180/pi, and get 30 and 60! Woot! Progress!

However, that's just the slope-angle. I need the actual slope (you know, the m in y=mx+b). So, I turn now to TOA, and since I have the slope-angle in question, this should be easier (since Excel wants everything in radians, I first have to convert my 30 degrees into radians (0.523599), and then take the tangent (0.577351) to get the slope of the hypotenuse (assuming that the adjacent angle is horizontal).

Then, I noticed that I could have simplified the whole thing by not transforming from radians into degrees and back again as well as combing the two processes into a single Excel command:

slope = tangent (arcsine (opposite/hypotenuse))

Of course, this only works when the "adjacent" side is assumed to be horizontal, which -- if one is doing surveying properly -- we can take the assumption as being true.

I'm sure that I will likely have to remember how to do differentiation and integration at some point in the future. And I did learn how to do it, and I was actually decent at doing it by hand at one point. However, that was approaching 10 years ago...

... after all, isn't that the purpose behind computers and mathematical software?

UPDATE: It's true that I could have just as easily gone for the a^2 + b^2 = c^2 right-angle association, knowing what a and c were, and using these to find b, and then taking the slope by using a/b... But that would be too simple! LOL.