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**Example research essay topic: Trigonometry** - 852 words

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Trigonometry is based on the study of right-angled triangles. This is a form of geometry that developed out of the study of the stars. It is also concerned with the relationships that exist between the sides and angles of triangles, and their measurement. Trigonometry comes in two forms, spherical and plane. Spherical is the study of curved triangles lying on the surface of a hypothetical sphere, which is mainly used by astronomers and navigators.

Plane is the study of more variety and is mainly focused on by many books on trigonometry. In trigonometry, the ratio between any two sides of a right-angled triangle is given as a function of angles within the triangle. These ratios are called trigonometric functions. Even though there are six types of trigonometric functions, three of them are commonly used ratios are sine, cosine, and tangent. The other three are cotangent, secant, and consecant.

Here is an example of how trigonometric functions of a general angle work. The angle, as shown in the diagram, is measured by usually showing the Greek letter theta, 0. The sine is the ratio of o, which is the length of the side opposite 0, to h, which is the length of the hypotenuse. The cosine is the ratio of a, the length of the side adjacent to 0, to h, the length of the hypotenuse. The tangent is the ratio of o, the length of the side opposite 0, to a, the length of the side adjacent to 0. sin 0 = o/h cos 0 = a/h tan 0 = o/a Trigonometry is considered to be an important component within every mathematics course, while its rules and formulae, the plural form of formula, has applications in subjects ranging from engineering to navigation.

An example of it being used for engineering is the Egyptians using it for building their pyramids. The functions of an acute angle that deal with any right triangle, it will be convenient to denote the vertices, the point at which the sides of an angle intersect, as A, B, and C with C the vertex of the right angle, to denote the angles of the triangles as A, B, and C = 90 , and to denote the sides opposite the angles as a, b, and c. With respect to angle A, a will be called the opposite side and b will be called the adjacent side; with respect to angle B, b will be called the opposite side and a will be the adjacent side. Side c will always be called the hypotenuse. This is the formula for the first figure drawn.

But if the right triangle is placed in a coordinate system so that angle A is in standard position, the point B on the terminal side of angle A has coordinates (b, a), and the distance c= a + b, then the trigonometric functions of angle A may be defined in terms of the sides of the right triangle. There are also laws for three of the six trigonometric functions. The law of cosines is that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. The law of sines are that the sides of a triangle are proportional to the sines of their opposite angles. The law of tangents is that the difference between any two sides of a triangle is to their sum as the tangent of half the difference between there opposite angles is to the tangent of half their sum.

An easy way to remember how the trigonometric ratios work is by the term SOHCAHTOA. This means Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent. By writing this report on trigonometry, I have learned that it is used for many practical applications, such as building, civil engineering, and navigation. I also learned that it can be used in situations where measurements cannot be made physically- for instance, finding the distance to a star or to an island. Over time, the subject of trigonometry has evolved from theorems on the ratios of the sides of triangles to helping astronomers figure out how far we are from other parts of the universe.

1. A train is moving at the rate 8mi/h along a piece of circular track of radius 2500 ft. Through what angle does it turn in 1 min.? Now I will solve it. Since 8mi/h= 8(5280)/60 ft/ min= 704 ft/min, the train passes over an arc of length s = 704 ft in 1 min. So then 0 = s/r = 704/2500 = 0.2816 rad or 16.13 .

2. Find the trigonometric functions of the acute angles of the right triangle ABC, given that b = 24 and c = 25. Bibliography: Works Cited 1. Mathematics Made Simple. Sperling, Abraham, Ph.D., and Stuart, Monroe.

Bantam Doubleday Dell Publishing Group: New York, 1991. 2. High School Review: Math II. French, Douglas. Random House, INC.: New York, 1998. 3.

Mathematics. Pascoe, L.C. NTC Publishing Group: Illinois, 1983. 4. How Math Works. Vorderman, Carol.

The Readers Digest Association, Inc.: New York, 1996. 5. Schaums Outlines of Theory and Problems of Trigonometry: Third Edition. Moyer, Robert E. and Ayres, Frank, Jr. McGraw-Hill, 1999.

Related: trigonometry, over time, civil engineering, random house, diagram

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