Trigonometry - Theory
Trigonometry - Theory |
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Trigonometry is a very important branch of Mathematics. Within the framework of Geometry, this tool makes easy knowing a lot of data about the angles and sides of triangles. Its name, from ancient Greek, means measurement of angles.
Let's consider a standard triangle. We can see that it has three angles, which will be named α, β and γ. Usually, the angles of a triangle are designated by these Greek letters, although it's also possible to use the letters A, B and C.
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A fundamental property of triangles is that the sum of the angles is always equal to 180º, as you can see in the image shown in the right. |
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Another characteristic of triangles (actually, of every geometric shape), caused by Thales Theorem, is that, if we change their size, the lengths are changed, whereas angles remain exactly as they were before.
This way, by knowing sides and angles of a triangle, we can easily find out angles of every triangle whose sides are proportional to the original.
Since a triangle has three sides, we must always know the three lenghts, because in the world of triangles,having two sides a and b with a common angle α is not the same as having two sides a and b with a common angle β. But, what if they were rectangle triangles, that is to say, with a right angle? All this would be quite easier. |
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For a beggining, we can consider that the two angles that are different to the right one are complementary. So, in order to know all of the angles of the triangle, we only have to know one angle. Then, we can assume that all of the triangles with a common angle will be proportional among themselves Then, as relationships between equal triangles are equal, we could stablish directly relationships between angles and sides of the triangle (cathetus and hypotenuse). |
For this task, trigonometric functions were created. They are a set of relationships between two sides for a particular angle of a right angle. Let's see them in more detail:
Sine (sin): the sine of an angle α is the ratio of the length of the opposite cathetus to the lenght of the hypotenuse. Cosine (cos): it is the quotient between the adjacent cathetus and the hypotenuse. Tangent (tan): it is the ratio of the opposite cathetus to the adjacent one. We also have the reciprocal functions corresponding to the original ones. Cosecant (csc): 1/sine, or the quotient between the hypotenuse and the opposite cathetus. Secant (sec): 1/cosine, or the ratio of the hypotenuse to the opposite cathetus. Cotangent (cot): 1/tangent, or the ratio of the adjacent cathetus to the opposite one. |
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Hay que aclarar que las razones tigonométricas son únicamente un cociente, una proporción. No tienen unidades propias, ya que no son una magnitud.
We need to clarify that trigonometric functions are only a quotient, a ratio. The have not an unit of measurement, since the functions are not a magnitude.
En muchas ocasiones sólo conoceremos una razón trigonométrica de nuestro triángulo. Para esas ocasiones son muy útiles las identidades trigonométricas, unas fórmulas que relacionan diversas razones.
We will often know just one trigonometric function of a triangle. In these cases, trigonometric identities (equalities that involve trigonometric functions) are very useful.
By using the calculator, it is possible to find out the trigonometric functions of every angle. For this mission, we have developed a small Flash animation that explains how to do this, and the other way around: calculate the angles while knowing their trigonometric functions.
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