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What Is Trigonometry?We learn from The Words of Mathematics:
In the ancient sense, trigonometry defines relations between elements of a triangle. In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line segments may serve as the sides of a triangle. They do iff they satisfy the triangle inequality, or rather three triangle inequalities. Not any three angles may be the angles of a triangle. In Euclidean geometry, the three angles of a triangle add up to a straight angle. These requirements impose limitations on the manner in which the relations between the elements are defined. Those limitations can be lifted if the relations are extended according to certain rules which occur naturally under the circular approach. If sides a, b, c of a triangle lie opposite angles α, beta;, γ then it is known from geometry that, say, There is an awful amount of trigonometric identities. The most basic one is the Pythagorean theorem expressed with sine and cosine:
Then there are double argument formulas:
and, more general, sum and difference formulas:
And, of course, no list of trigonometric relations could be complete unless the Laws of Cosines and Sines are mentioned. Trigonometry is a methodology for finding some unknown elements of a triangle (or other geometric shapes) provided the data includes a sufficient amount of linear and angular measurements to define a shape uniquely. For example, two sides a and b of a triangle and the angle they include define the triangle uniquely. The third side c can then be found from the Law of Cosines while the angles α and β are determined from the Law of Sines. The latter can be used to find the circumradius. The area of the triangle can be found from References
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