Sunday, December 22, 2019

Physics Physics Of Trigonometry - 1102 Words

The more I study trigonometry, the more it seems like mathematical magic. The relationships between angles and sides that have been uncovered and rewritten as seemingly simple identities are definitely more than meets the eye. I am especially drawn to their all-inclusive nature. It does not matter what the side lengths or angle measurements of the triangle are; the laws of trigonometry are always applicable and accurate. Needless to say, I was quite intrigued when I came across something known as Morley’s Theorem which reads, â€Å"The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle† (Bogomolny). What interests me most is the seemingly unlikely all-encompassing nature of such†¦show more content†¦Figure 2 The summation of all of the angle measures in a triangle is equal to 180 º which can be converted to equal Ï€ radians. Thus I deduced that ∠ A+∠ B+∠ C=Ï€ 3ÃŽ ±+3ÃŽ ²+3ÃŽ ³=Ï€ ÃŽ ±+ÃŽ ²+ÃŽ ³=Ï€/3 The first step in this proof is to find expressions for the side lengths AB, BC, and CA; however, that is a difficult task to accomplish since I am only using variables and no concrete measurements. In order to overcome this challenge, I circumscribed a circle about the triangle with a center O so that the triangle only touches the circle at its three vertices A, B, and C. For the sake of simplicity, I assume that the radius is 1, so that I am able to find the side lengths of the triangle in terms of ÃŽ ±, ÃŽ ², and ÃŽ ³ by applying rules of basic geometry and the cosine rule. According to the Central Angle Theorem, â€Å"the central angle subtended by two points on a circle is twice the inscribed angle subtended by those points† (â€Å"Central Angle†). Thus, I know that if m(∠ CAB)=3ÃŽ ±, then m(∠ COB)=6ÃŽ ± as illustrated by figure 3. Figure 3 Since BO and CO are radii in the circle above, they must be equal to 1. Now that I have a value for the measure of ∠ COB, I can use the law of cosines to find the length of side BC because I have measurements for an angle and both of its adjacent sides. ã€â€"(BC)ã€â€"^2= ã€â€"(BO)ã€â€"^2+ã€â€"(CO)ã€â€"^2-2(BO)(CO)cos⠁ ¡(m∠ (BOA)) ã€â€"BCã€â€"^2=1+1-2∙cos⠁ ¡(6ÃŽ ±) ã€â€"BCã€â€"^2=2-2∙cos⠁ ¡(6ÃŽ ±)Show MoreRelatedPi And The Real World995 Words   |  4 Pagesthrough college know of pi and its multiple applications. It has been used in physics, as well as in geometry. Students will even use it in trigonometry when they are doing sine waves. Students need to see how necessary pi is in mathematics and in the real world. Although it may seem pointless to some students now pi will help in numerous career fields as well as in more accelerated classes like calculus A-B or AP physics. 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