## Friday, January 18, 2013

### On trigonometry and measuring trees pt.1

I love trigonometry. It is that weird kind of complicated love one has for a subject that its tough enough for you to be glad you left it behind, but cool enough to pursue from time to time.

I can still recite the Pythagorean theorem by heart: The square of the sum of the "legs" of a triangle equals the square of the hypotenuse.
Before my review I had a hazy recollection of sines, cosines and tangents. I remembered their graphs being waves and that they were interrelated somehow.

What I did remember was that in high school we measured the height of our school using trigonometry. On that assigment my calculations were off enough to get less than 90% of the points. I had not thought about that in years, but I know it bothered me back then, since I knew no one had gone and measured that wall directly...so how could I know the teacher was right? The fact that the teacher, Mr. Quintero, had been doing this for years, and that countless of students had done this calculation accurately before me did not enter my mind then. I just focused on trying to prove I could be right.

After all that history, let me surmise a little of what I have reviewed of trigonometry. Trigonometry is the study of triangles, specifically those with one right (90 deg) angle. The relationships of the sides of that triangle gives us the Pythagorean theorem.

From the relationships of the sides and angles of those right triangles we get the trig functions sine, cosine and tangent.
Therefore, following the rules in the picture, if we know any two sides, or any side and one angle other than the 90% one. We can solve all other angles and sides. At first glance this information sounds baffling, but trigonometry is one of those fields in mathematics that is immensely practical.
To know that as long as you can draw an accurate right triangle, the relationships of its sides remain the same so mater how big or small that triangle is, is simply amazing. In fact, it is one of those constants in nature, alongside Pi, that makes mathematics such a transcendental discipline. One that has spawn more than a few religions, as Pythagorans might attest to.

During my review I ran into at least 2 exercises of measuring trees. They would give me the distance to the tree and the angle of elevation to the top of the tree. With that I must be able to calculate the height. But how?

Well, if you look at the picture of the triangle above and compare it to the data. You can conclude that I have been given the value of the adjacent side (the lenght between me and the tree) and the angle θ (the elevation from my point of view to the top branch of the tree). What I need to find out then, is the lenght of the opposite side (height of the tree). Do any of the formulas above have the two given variables and the unknown variable?

Yes, tanθ. We know that tanθ = opposite/adjacent. If we need to find the height of the opposite side we rearrange the formula to opposite = tanθ(adjacent). Which means that the lenght of the opposite side equals the tangent of the degree corresponding to that side times the lenght of the adjacent side to that angle.

Let's imagine this example: We are standing 30 feet away from a tree. We lie flat on our stomach, take a protractor and use it to see the topmost branch of the tree. When we do so, the angle of inclination is given as 53.13 degrees. How tall is the tree?
We just derived that opposite = tanθ(adjacent), substituting our given data we get. Opposite=tan(53.13)(30 feet) or opposite=(1.33)(30 feet). That would make our tree approximately 39.9 feet tall. Cool, right?
Well, not content with the excercises alone I decided to do a real world measurement. However, considering my past experience measuring the height of my high school and not having any direct measurement for it, I decided to try it on a tree I could measure. So, I used our Christmas tree. I'll tell you all about it in my next post.