To start the “Measuring Your World” project we had to prove why the Pythagorean theorem worked. To do this we needed to know the equation of a^2+b^2=c^2. We started this off with “proof by rigs” where we were given two squares with different shapes in them. One had the shaded area of A and B the other had C. We had to find if these were even to one another (This is the second photo). Once we split the shapes into similar right triangles we just found the areas of the squares, knowing that's just the variable squared.
Now that we knew how to work with right triangles lengths we started working with them on a Cartesian coordinate plane. In the past we'd usually only worked with rise over run on planes, but not too many shapes. On each of these sheets we knew the hypotenuse of the triangles but had to find the missing lengths, while knowing all of the angles. We used reflection over lines of symmetry to make larger and equal shapes, or even made quadrilaterals to find the area and then overall lengths of the shapes. This was all to prove the distance formula ( √(x2−x1)2+(y2−y1)2) on what we call the unit circle (The Fourth photo) The unit circle is a circle with a radius of 1 centered at the origin. With unit circle you can measure right triangles and find multiple special right triangles. To understand unit circle symmetry you can calculate unknown points by reflecting known points onto their position. This means that each will be the inverse when reflecting over each axis. While using the unit circle we can use the right triangles we already know and derive SOH CAH TOA, or Sine=Opposite/Hypotenuse, Cosine= Adjacent/Hypotenuse, Tangent=Opposite/Adjacent. Since we know the multiple side lengths and angles we can set up these equations and solve for the variables as necessary.
After learning the basics of SOH CAH TOA we learned about ArcSine, ArcCosine, and ArcTangent. These all represent the inverse of sine, cosine, and tangent, knowing this we understand that ArcCosine is: angle theta=cos^-1 same goes for ArcSine and ArcTangent. For example the ArcSine is the exact opposite of the sine functions. sin30=0.5 while the ArcSine of 0.5=30, this goes for ArcCosine and ArcTangent. We then worked on the Mount Everest Problem which worked with triangles that weren't right triangles which meant the Pythagorean Theorem wasn't applicable here. But we could look for right triangles by dropping a perpendicular (a line to make a right angle going the shortest distance from one side to the other) this made a right triangle that we could work with. Next we had to work with the Law of Sines two, or three, of the points given and one of the side lengths to solve for the missing sides. we worked out the law of sines is (Capitol letters are angles while lower case are side lengths) sinB/b = sinC/c = sinA/a. An example is sin100/86=sin50/a, then we simplify to a=86 x sin50/sin100. The law of cosines is utilized when we know two lengths and one angle between them, this is written as c2 = a2 + b2 – 2ab cos C. Every one of these equations can be connected to any issue with or without a right triangle that requires searching for a length or height. An example for this is: c^2= 75^2 + 15^2 - 2 x 75 x 15 x cos20, simplified to c= 61.12.
Part 2
I chose to measure the easiest place to access, our classroom. The classroom worked for this project because it is separated into two areas, the entrance or smaller area, and the workspace where we all are. I chose to figure out the larger one first because of it’s high ceilings and long walls.
The variables I needed to figure out were, the height of the larger room, the length of that room, and the angle at which the hypotenuse would be at. The formulas that I used were TOA or Tangent= Opposite/Adjacent, and Length*Width*height=Volume. I needed TOA because of the height of the room which is well over my height of 180cm. I created the room into a triangle where I knew the base length and the angle of the room but not the height. See the images below for the process. My measurements were all eventually converted to cm and added together.
Overall I don’t think I faced any challenges that stopped me from working. All of the math I used to find the answer was simple and didn’t need explanation only the application to this real world place was initially difficult. The first habit of a mathematician I used was staying organized, this was extremely useful in the end. At the beginning of the mini project I was writing all of my notes out of order and in only batches of numbers I needed at the time. Then I started writing all my material in order and had all of the steps easily locatable. The next habit of a mathematician I used was being systematic, this ties in with the last habit because I started to be systematic when I became organized. Because I wasn’t in a group this project I found it easy to decide on what to do and found the project overall interesting because we were truly applying what we learned.
Beow are the photos of the room I was measuring.