## DP Update #1

We learned many new things throughout this project. Starting off we learned about the Pythagorean Theorem, A2+B2=C2. In this theorem C represents the hypotenuse while A and B Represent the remaining two sides of the triangle. You can use this equation to find out the missing sides of a triangle with the known sides. We worked on a sheet known as "Proof By Rugs". On this sheet we took a rectangle and divided it up into right triangles which we then found the lengths of the triangles sides using the Pythagorean Theorem. We then went onto deriving the distance formula. The Distance Formula is (X2-X1)2+(Y2-Y1). As we moved forward the next step was using the Distance Formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane also known as a Unit Circle. With two points we were able to derive the distance formula by plugging in given points into x2+y2=4 and solving for the distance between them. Next was the unit circle, with a radius of 1 it allowed us to have a good reference point from then on forward. The circle had measurements for points that helped us move forward and understand where we had to be for our equations. We also had to find points on the unit circle at 30,45, and 60 Degrees. When you look at these angles the 30 and 60 are the same but just flipped. If you draw a triangle going through all three of these points, you can see that the angles recreate themselves along the x axis. This was something that helped me further understand that many of these things connect in many different ways. The symmetry of a circle was used to find the remaining points on the unit circle. This was relatively easy because of the points we had already found. Because these points were on a circle this meant that each quadrant would only change the points very little because they would all be similar. Either flipped or negative or something similar to the original. Also throughout all this we used Sine, Cosine, and Tangent. These three equations or actions were relatively easy for me to grasp. In my perspective I viewed them as almost a new Multiplication and Division for these scenarios with triangles and the Pythagorean Theorem. To define sine and cosine we took a triangle in the Unit Circle where we then tried to find one of the 2 lengths of the 3 sided triangle, with all the angles given to us. From there we used sine and cosine to solve the triangle. Defining the Tangent function, by dividing the opposite over adjacent the tangent function is finding the hypotenuse or angle when you only have the opposite or adjacent side lengths, in which you can find the missing hypotenuse or angle. Next we used similarity and proportion to find our cosine, sine, and tangent functions. If we look at the information that's given (angles, lengths, etc.), you can determine the cosine, sine, and tangent through SOH-CAH-TOA. SOH, which is sine = opposite over hypotenuse, CAH, which is cosine = adjacent over hypotenuse, and TOA, which is tangent = opposite over adjacent. These were a little hard to grasp but after I got the hang of them I was able to use them successfully. Arcsine, Arccosine, and Arctangent were when the -1 is in front of the sine, cosine, or tangent functions. This was just the function but negative. This was also very easy to grasp as it was almost the same thing just negative. Next was using the Mount Everest problem to discover the Law of Sines. We were given background on Mount Everest and information on how it was pinpointed. With this we all tried to find the location of Mount Everest. The key to finding its location was using the Law of Sines. This was the fastest way to get the location of Mount Everest. Our last step in all this was learning to derive the law off sine's and law of cosines. Learning Law of sine's first really helped me understand Law of cosines which came second. Overall this was all a great experience learning all this new information.