Why do we have both sine and cosine?

[Maurice7510]

This might seem like a really basic question that one might cover in gr 9 or 10 but instead my friend and I were discussing it now, when he just got his degree and I'm a credit away from mine: why on Earth are there both sine and cosine functions when simply one would do? Either can be expressed in terms of the other with a phase shift of π/2 and there weren't any identities we could think of that would necessitate an entirely new function to express that. My only thought was that maybe standard sine and cosine functions weren't originally posed as themselves, instead having been derived using complex exponentials -- the +/- between the terms could warrant the use of separate functions for a notational convenience. The other thought I had was that maybe they weren't posed first, and were derived from their hyperbolic analogs.
While neither of my suggestions seem likely, we couldn't think of any other reason that we would need to define two functions instead of one function of different arguments.
If anyone has any insight I'd love to hear it. Thanks!

 

[mathman]

Math students first encounter them in elementary trig., where it is simpler to define them separately, as well as tan. Later on one learns sec, csc, and ctn. Personally I find the last three (also tan) as annoying, so working with trig functions I prefer to reduce everything to sin and cos.

Complex exponential and hyperbolic functions come much later.

 

[SteamKing]

This might seem like a really basic question that one might cover in gr 9 or 10 but instead my friend and I were discussing it now, when he just got his degree and I'm a credit away from mine: why on Earth are there both sine and cosine functions when simply one would do? Either can be expressed in terms of the other with a phase shift of π/2 and there weren't any identities we could think of that would necessitate an entirely new function to express that. My only thought was that maybe standard sine and cosine functions weren't originally posed as themselves, instead having been derived using complex exponentials -- the +/- between the terms could warrant the use of separate functions for a notational convenience. The other thought I had was that maybe they weren't posed first, and were derived from their hyperbolic analogs.
While neither of my suggestions seem likely, we couldn't think of any other reason that we would need to define two functions instead of one function of different arguments.
If anyone has any insight I'd love to hear it. Thanks!

This is how the 6 trigonometric functions were originally defined:

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Each if the three functions in the first row, sine, cosine, and tangent, has a complementary function defined on the second row, which function is just the multiplicative inverse; for example, cosecant A = 1 / sine A. Sine and cosine are ratios of the opposite and adjacent sides of the triangle with the hypotenuse, and yes, they are connected by the well-known identity sin² A + cos² A = 1. The exponential definition of the trig functions came many centuries after the geometrical definition, and the familiar trig identities are probably easier to derive using the geometry of the triangle than trying to manipulate an exponential, at least for teens taking trig for the first time.

Sometimes, most of the utility of defining similar functions is that each is convenient to use in certain situations.

 

[Mark44]

The coordinates of a point on the unit circle are $(\cos(\theta), \sin(\theta))$. This is a lot more convenent than, say, writing $(\cos(\theta), \cos(\pi/2 - \theta))$.

 

[leetramp]

I would say that sine, cosine, and tangent are just ratios of sides of a triangle, thus we have three tables to look up on (oops, I'm showing my age). It's only when you get to the unit circle that you run into looking at "why three?"

 

[SteamKing]

I would say that sine, cosine, and tangent are just ratios of sides of a triangle, thus we have three tables to look up on (oops, I'm showing my age). It's only when you get to the unit circle that you run into looking at "why three?"

There are several more trig functions which have fallen out of common use, but which once appeared in trig tables:

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

These other trig functions are defined in terms of the usual sine, cosine, and tangent, but while these functions were once used for surveying, navigation, and astronomy, they seem to have dropped out of regular use. In their heyday, though, they were quite convenient for doing calculations with the aid of trig tables.

 

[Delta2]

Well steamking's explanation and diagram show that sine and cosine at a first glance seem to be two totally different things (one is opposite side/hypotenuse, the other is adjacent side/hypotenuse)

But then again one can see easily , that the sine of the one acute angle $\phi$ of the orthogonal triangle equals to the cosine of the other acute angle which is $90-\phi$,

So i guess yes we could ve lived and evolve trigonometry with just sine or cosine.Maybe for intuitive reasons it seems more straightforward to say $cos\phi$ than $sin(90-\phi)$.