Measuring Your World
My class began this project by looking at a well known formula called the Pythagoras Theorem, a² + b² = c². (C) is the hypotenuse, which is always the longest side.The use of these lengths are to find the missing length of any right triangle by plugging in known side lengths with equations. There are multiple different ways to prove the Pythagorean Theorem true, but the way we were taught is achieved by taking a triangle divided into two smaller right triangles. We were able to show this proof by disecting the triangle into 3 similar parts and prove that a² + b² = c² still stands true. Next to make a proof of the distance formula, we can use a² + b² = c² and re label our triangle with the coordinates, a and b.
My class began this project by looking at a well known formula called the Pythagoras Theorem, a² + b² = c². (C) is the hypotenuse, which is always the longest side.The use of these lengths are to find the missing length of any right triangle by plugging in known side lengths with equations. There are multiple different ways to prove the Pythagorean Theorem true, but the way we were taught is achieved by taking a triangle divided into two smaller right triangles. We were able to show this proof by disecting the triangle into 3 similar parts and prove that a² + b² = c² still stands true. Next to make a proof of the distance formula, we can use a² + b² = c² and re label our triangle with the coordinates, a and b.
We can then use this to obtain the equation of a circle centered at the origin of a Cartesian coordinate plane, or the center of a graph, also known as a unit circle. The circle has a radius of 1. All points of any circle, including the unit circle are equidistant from a given point. A point such as this is called the Locus. We have proved all circles are similar because they are regular. This means they can be overlapped on their origin and dilated to be the same size. The equation x² + y² = r² which follows a similar form to both the pythagorean theorem and the distance formula can be used to find points on the unit circle. Using the unit circle we can create special triangles with angles of 30 degrees, 45 degrees and 60 degrees. We use the formula x² + y² = 1 to find the 45 degrees- 45 degrees angle triangle. in this triangle (x) and (y) are equal.
When solving for 30 degrees, it is possible to reflect the triangle over the x-axis creating an equilateral triangle after being reflected. All sides become 1 and y is easily divided by 2 to create the side length of 1/2. This gives us enough information to finish solving.
Lastly, if we are solving for a triangle at 60 degrees, we can solve this very similarly, reflecting into an equilateral triangle, then finding the length of (x) which should be 1/2 the solving for the rest with the Pythagorus Theorem.
Understanding the unit circle symmetry, we can calculate the location of unknown points by reflecting know points onto their position. Each will be the inverse when reflecting over each axis.
Lastly, if we are solving for a triangle at 60 degrees, we can solve this very similarly, reflecting into an equilateral triangle, then finding the length of (x) which should be 1/2 the solving for the rest with the Pythagorus Theorem.
Understanding the unit circle symmetry, we can calculate the location of unknown points by reflecting know points onto their position. Each will be the inverse when reflecting over each axis.
Using the unit circle, we can make a proof of tangent by reflecting it upon the x-axis and it will overlap perfectly, proving it is 90 degrees and perpendicular.
We learned about each of these trigonometric formulas by using a shape as simple as a right triangle. To start off, labeling the triangle in relation to where the delta angle is located helps to show which lengths you will working with, the sides are labeled: opposite, adjacent, and hypotenuse. We learned about sine by understanding that the equation S=O/H (or sine equals opposite divided by adjacent) If we are looking for a y-coordinate on the unit circle we can use sine to solve. If we are instead looking for the intersection of a point on the x-axis on the unit circle, we can use cosine to solve. The equation used to solve for cosine is written as C=A/H. Lastly, if you are looking for tangent the proper equation is written as T=O/A. We learned about these key trigonometric terms by dissecting the unit circle into triangles that allowed us to look deeper into the intercepts and angles to solve for a specified point.
The next set of terms we covered are ArcSine, ArcCosine, ArcTangent. These can each be derived by finding the inverse of either sine, cosine, or tangent. Knowing that the inverse of something will always be its negative so we learned this concept by understanding that ArcCosine is: angle theta=cos^-1 and so on for sine and tangent. We learned about the Law of Sine by working with triangles that weren't right triangles, meaning that the pythagorean theorem could not be applied to them. One of the problems we solved using this method was the Mount Everest problem, which we split into to right triangles by creating a perpendicular lines and then solving for the missing length. We were able to use this method because we knew two of the angles and one of the side lengths given. When written out the law of sines is sinB/b = sinC/c = sinA/a. The law of cosines is used when we know two lengths and one angle between them, this is written as c² = a² + b² – 2ab cos θ. These formulas can be used with or without a right triangle looking for length and height. I personally found these to be quite straight forward, just plug in the values and solve.
Measuring Your World 2
The object my partner and I chose to measure, was the largest of the Pyramids of Giza. This was an obvious choice for us because we knew we had to find the area, Volume, and a trigonometry try and involve trigonometry in a calculation. We were able to get approximations of the measurement, such as length and Height, of the pyramid from accredited websites. Sadly we weren't able to travel all the way to Egypt, so we had to rely on websites. A simplification we made when measuring the areas and volume of the pyramid was the sides. The sides are not flat, but we calculated our data overlooking that. Technically our calculations are missing the individual blocks sticking our of the pyramid.
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Reflection
I struggled a little through this project, mostly because there were a lot of other things happening in my life in and out of school. I remember having many due dates around this time and I was having trouble focusing at home. besides all of that I still strived to complete the project, as I should. I was a little flustered the morning before my presentation and when I went up to present and I made a few mistakes. I felt extremely embarrassed during my mistakes in the presentation, and it made it seem that I hadn't put effort into my work. I think the Habit of a Mathematician I used the most was make a list, or be organized. I showed these because when I started working on this I had a list of things that I needed to do. Every time I completed a part of the list I crossed it off. Unfortunately for myself this may have caused me to rush. I could have fixed that if I worked backwards and checked my work. One thing I would have wanted to to in this project was start working on it earlier. I felt like this project was a little rushed and I could have started working on it as soon as possible to get a head start.
I struggled a little through this project, mostly because there were a lot of other things happening in my life in and out of school. I remember having many due dates around this time and I was having trouble focusing at home. besides all of that I still strived to complete the project, as I should. I was a little flustered the morning before my presentation and when I went up to present and I made a few mistakes. I felt extremely embarrassed during my mistakes in the presentation, and it made it seem that I hadn't put effort into my work. I think the Habit of a Mathematician I used the most was make a list, or be organized. I showed these because when I started working on this I had a list of things that I needed to do. Every time I completed a part of the list I crossed it off. Unfortunately for myself this may have caused me to rush. I could have fixed that if I worked backwards and checked my work. One thing I would have wanted to to in this project was start working on it earlier. I felt like this project was a little rushed and I could have started working on it as soon as possible to get a head start.