 My Math Forum Origin Of Trigonometry Functions Not In Right Triangles
 User Name Remember Me? Password

 Trigonometry Trigonometry Math Forum

 June 28th, 2016, 03:50 PM #1 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 637 Thanks: 85 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? June 28th, 2016, 04:03 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,616 Thanks: 2605 Math Focus: Mainly analysis and algebra 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. Thanks from EvanJ June 28th, 2016, 06:03 PM   #3
Senior Member

Joined: May 2016
From: USA

Posts: 1,310
Thanks: 550

Quote:
 Originally Posted by EvanJ 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. Tags functions, origin, triangles, trigonometry Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mann001 Trigonometry 2 November 20th, 2015 12:02 AM vincent Calculus 7 October 17th, 2014 09:42 AM danield3 Algebra 2 April 28th, 2010 06:18 PM CHRISTOPHERHENRY Linear Algebra 3 March 30th, 2009 07:18 AM paulgarrard501 Number Theory 0 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      