Question

New to real math, trying to understand trigonometric functions like sine, cosine, etc.

I understand that sin = opp/hyp, but I don't understand why sine has such broad applicability. Why, for instance, is sine involved in problems that are fundamentally non-geometric?

Answer 1

continued:
I have asked an associate about this and his answer was basically "I don't know", but he postulated that sine is relevant in calculus, etc., because the objects being operated upon are encoded into a geometric form. That is, math that involves time uses sine because the time is represented as a triangle/graph, for purposes of calculation. Another associate found this explanation dubious.

Answer 2

You can use sine and other trig functions to work with things like, for instance, frequency. In fact, a lot of applied math deals with them.

Answer 3

OK, but that's not helpful really. I understand that it is widely used. I want to know *why* it is used to calculate things that are not intuitively represented as triangles. Sine seems hard defined as opp/hyp, so if there is no triangle involved, where does the sine come from?

Answer 4

There must be a triangle, or else there can be no sine, because the argument of sine must be an angle (or radians, but that's another story.)

Answer 5

If there isn't a triangle, then if there's sine or another trig function there then you can make a triangle from it.

The point is, it's a helpful way to relate two terms as a function of a third, in the same way that PV=nRT (ideal gas law) is useful in that it relates things like number of moles, pressure, volume, and temperature.

Answer 6

triangles are everywhere

Answer 7

maybe you are referring to the fact that sinus is analytically defined as a taylor series and therefore can be used as a lot of approximations. as a physicisit i have to say that every time something periodically comes up i have to use sinus...

Answer 8

Hello ,
My students often ask this question - and IT IS A GREAT AND UNUSUALLY DEEP FACT of science the Importance and ubiwuitious appearance of Trigonometric functions. Hera are two CENTRAL REASONS I KNOW:

Answer 9

Reason 1fundamental
Both Sinus and Cosinus are the simplest basic solutions of the vibration/waves/oscillation/resonance equations. One such (not the only one !) is

y'' - y = 0

Answer 10

shouldn't it be y''+y=0?

Answer 11

b/c spin is fundamental
/thread

Answer 12

y''+y = 0 (thanks)
Reason 2 Fundamental: Discovered by Jean Batiste Joseph Fourier:
ALMOST EVERY FUNCTION, EVERY TIME DEPENDENCE, EVERY SPACE DEPENDENCE in nature can be represented by SUMS OF SINE functions and COSINE FUNCTIONS of different frequencies.
One can say in paraphrase:
Everything in the universe is sum of waves. Simplest wave is Cosinus and Sinus

Answer 13

http://en.wikipedia.org/wiki/Fourier_analysis

Answer 14

http://en.wikipedia.org/wiki/Harmonic_analysis

Answer 15

http://en.wikipedia.org/wiki/Joseph_Fourier

Answer 16

Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain. This justifies their use in branches such diverse as image processing, heat conduction and automatic control.

Answer 17

http://en.wikipedia.org/wiki/Heat_equation

Answer 18

Great Question !!
Einstein and Bohr repeatedly explained that the ability to ask the most naive questions leads to the most deep answers about Nature.

Answer 19

Thanks for the answers everyone. I will follow up on the links Mikael posted. I guess the tl;dr answer *is* that everything is encoded as a triangle for calculation.

Answer 20

I would not say everything is a triangle... rather, right triangles led to the idea of sin, cos, tan (or secant in the Greek times). These trig functions were extended to the unit circle (angles 0 to 360) and then to all angles. In turns out this extension was very useful for describing periodic things, for defining polar coordinates (which leads to representing complex numbers as sin/cos). Then Fourier showed that useful functions can be represented as the linear combination of sin/cos...
So not triangles, but periodicity is the secret to its usefulness....

Answer 21

You guys are awesome.