This unit began in January 16, 2018 where we explored measurement through one dimension to three dimension studying volume. We were also able to prove Pythagorean Theorem, distance formula, law of: sine and cosine. Throughout this unit were were given twenty-four worksheets to complete which all had practice to what we were learning in class. Further on, I will be showing how and what I was able to learn in this unit.
Trigonometric Formulas:
Sine We were given a worksheet with simple trig practice in which we had to find missing lengths of triangles. Our teacher began to explain when we needed to use sine and what number to use. We used cosine and sine, in order, to verify we used the Pythagorean theorem. When everyone in our class had finished solving this worksheet we came together and shared our answers with each other in our table groups and as a class. Cosine We learned cosine when we learned about sine, by a worksheet that our teacher handed out. Before handing the worksheets out out teacher was also able to explain how we would apply cosine to the problem. However, before learning about trigonometry, we began to prove Pythagorean Theorem that then lead us to understand trigonometry. Tangent We learned about tangent differently, we were given another worksheet in which we had to find the length of a tangent on a unit circle. Along with finding the tangent we were also able to prove that tangent equals to sine θ over cosine θ. We got to practice using sine, cosine, and tangent while trying to find missing sides of right triangles. Along with practicing we also learned about soh cah toa which was very helpful when determining which trig formula to use. ArcSine - ArcCosine - ArcTangent We learned about these trigonometric formulas through finding missing angles of right triangles. Personally, it was a bit challenging for me to determine which formula to use along with angles. However, we were always allowed to work with our peers to solve these problems which were always of great help. It allowed us to share our ideas on how to approach each problem. Law of Sines (ASA) Previous to this school year I already had practice with basic trigonometry yet I don't remember going over the law of sines. This was of great help to solve many problems and understand why we used this formula. I was able to understand that we use the law of sines when we are given two angles and a side length. Law of Cosines (SAS) Along with learning about the law of sines, I was able to learn to use the law of cosines. In order to use it, we needed to know two sides and an angle of a triangle. I found this law much easier than the law of sines essential because all I needed to do was plug in the numbers into the equation.
Steps in my learning:
As a class, we were able to prove the Pythagorean Theorem a which led us to derive the distance formula. From deriving the distance formula we were able to derive the equation of a circle centered at the origin of a Cartesian coordinate plane. Leading us to define a unit circle by finding points on at 30 degrees, 45 degrees, and 60 degrees. Next, we began to use the symmetry of a circle to find the remaining points on the unit circle learning to drop a perpendicular allowing us to create a right triangle. This helped define sine and cosine of the angle theta (θ). We learned about sine and cosine through worksheets that our math teacher was able to provide which included a diagram allowing us to visualize. We defined the tangent function with diagrams along with the direction of our teacher. Notes were a lot of help when learning about the tangent functions. All of the worksheets had questions or formulas we needed to prove, our teacher was able to walk through the proof and then we were given time to solve problems from we had just learned. Using similarity and proportions helped us derive the general trigonometric functions; sine, cosine, and tangent. More specifically triangles, we set up proportions when proving a formula. With the unit circle, we were able to define arcSine, arcCosine and arcTangent function by observing how the different point allowed us to solve for angles in a triangle. Our teacher gave us a problem that was not just a function but a word problem about the Mount Everest to discover the Law of Sines. He allowed us to work with our table groups yet it was really difficult when we needed to know about the Law of Sines. Our teacher helped us derive the Law of Sines and we were able to solve the Mount Everest problem with it. Since the Law of Sines and Cosine was new concept we were taught about how we can prove it and how to apply it to problems. Once we had an understanding of the Law of Sines and Cosines we were given problems to practice and review.
Part 2:
Intro: For the second part of the project, we were given the task to complete a project introduced to us. The project was to measure the volume of an object that we use everyday or an object of choice. A soccer ball was an object that we were encouraged to measure due to the level of difficulty. Another requirement we needed to use was a trigonometric function, an area formula and a volume formula along with a presentation. It was never specified that we need to have a slideshow presentation yet it was what most of my classmates chose to use. In this project we were allowed to work alone or in a group up to three members. Math: For the math portion of this project, we had to use a trigonometric function, an area formula, and a volume formula. Knowing this information my group was able to find a way in which we could include these functions and formulas. It was easy to do so because in order to find the volume we first needed to find the height and area of the pentagons and hexagons that were on the soccer ball. It was a somewhat lengthy process yet we were able to complete and finish on time. Reflection: Thought out this project I was able to take apart and put back together which is a habit of a mathematician that my group was able to use. If we were to this project again I would choose to do an object with a higher difficulty in order to challenge myself more. Overall, this project allowed us to practice our knowledge on measurement and apply it to there objects we measured.