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June 28th, 2016, 03:50 PM   #1
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Origin Of Trigonometry Functions Not In Right Triangles

In basic trigonometry, you learn:

sine = opposite/hypotenuse
cosine = adjacent/hypotenuse
tangent = opposite/adjacent

In an equilateral triangle, the ratio of any two sides is 1. 1^2 = 1. Obviously the sine and cosine of a 60 degree angle cannot be 1 each because it violates sine^2 + cosine^2 = 1. So how are trigonometry functions of angles not in right triangles determined?
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June 28th, 2016, 04:03 PM   #2
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Drop an altitude from one of the points (perpendicular to the opposite side of the triangle). You then have two right-angled triangles for which to calculate the trigonometric functions.
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June 28th, 2016, 06:03 PM   #3
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Quote:
Originally Posted by EvanJ View Post
In basic trigonometry, you learn:

sine = opposite/hypotenuse
cosine = adjacent/hypotenuse
tangent = opposite/adjacent

In an equilateral triangle, the ratio of any two sides is 1. 1^2 = 1. Obviously the sine and cosine of a 60 degree angle cannot be 1 each because it violates sine^2 + cosine^2 = 1. So how are trigonometry functions of angles not in right triangles determined?
The definition of sine and cosine in terms of hypotenuse obviously implies that the definitions are made with reference to a RIGHT triangle. An equilateral triangle is NOT a right triangle so I can see why you asked your question.

A trigon has three sides, just as a pentagon has five sides, a hexagon six sides, a heptagon seven sides, and an octagon eight sides. In other words, trigon is just a synonym of triangle. And "metry" has the meaning of measurement. So the name implies measurement of triangles generally. And trigonometry is useful for measuring all sorts of figures that are not triangles at all. So what's the explanation for going from right triangles to many other kinds of figures.

Well Archie gave you the answer. Any closed rectilinear figure of n > 3 sides can be decomposed into (n - 2) triangles, and every triangle can be decomposed into the sum of two right triangles.

The classical definition of sine and cosine is made in terms of right triangles, but the decomposition rules mean that the use of sines and cosines can be expanded far beyond right triangles.

Last edited by JeffM1; June 28th, 2016 at 06:06 PM.
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