Polymath Trigonometry
Trigonometry class training is for your local school, if you have not already done so. Herein, I only want to describe Trigonometry a bit, and then show some items I came up with in my quest of the regular polygon and circle relationship. However, it is very unlikely if you find, Trigonometry in most schools at high school or college level as depicted in this web page.
The main core of trigonometry is the right triangle. Shown in Fig.-22 as OPG. OP is the hypotenuse of this right triangle, and always has the value of one (1). GP is the perpendicular leg line of the triangle and is called sine. OG is the horizontal base leg line of the triangle and is called cosine. The angle that the hypotenuse and the base of the triangle, shown as o varies from 0 to 360 degrees. That is why there is a circle shown. (Normally most Trig math is accomplished using only 90 degrees).
Whenever the angle o is changed all the lines change length, except for the radii. One of which is as mentioned, the hypotenuse. This is why we compare all the line lengths to one (1)
Thus for any angle, there are a bunch of line lengths of specific size associated with the same angle.
What is Trigonometry good for? Geometric problems that only have an angle and one side of a right triangle can be solved. Thus we can tell how tall a tree is without climbing it. We can find out how far a boat is in the ocean without swimming out to it.
In the past I looked at the geometric layout shown in Fig.-22, and wondered about exchanging the circle for a square?
The result looks like Fig.23. Right away I noticed the hypotenuse would be changing length as the center angle changed. The sine and cosine were also different in length. But some other lines were similar. So I moved on and went with it... Refer to Fig.-24.
first chapter page prior. My poly radI changed the hypotenuse to what I call Cant. The two lines cr are coradii in accordance with my ius is shown as r.
The bottom line is: I worked out what I thought was a new type of trigonometry. It worked! I was pretty elated for a while. Then I conferred with a Professor at Gonzaga University, and he pointed me to data that showed others had done this before I.
But, I didn't know this when I did my work, so it still makes me feel pretty good anyway. I am not going to show all the formulae here, but I still retain the thought: I may still have a few items within this math that others did not. For example: I changed 360 degree to just 1. Thus all angles are in decimal fraction, and I went on to calculate out tables for the lines on a computer to 15 decimal places, and down to very small angles. It was interesting and a challenge. At the time I did the programming in double precision BASIC.
Is there any use for this type of Trigonometry? Maybe a couple of things. I did find that at small initial angles, sine is easier to physically plot on paper, since it grows more readily at small angles. It also showed me there was, and probably still is, math unknown to many of us.