Comments on: How many trig functions are there?

By: Timmo

Timmo — Wed, 08 Feb 2012 18:07:31 +0000

Of course, you don’t even need e or i to get the trig functions! You can define sine purely through its series expansion. Then, all you need is the reals.

By: Xah Lee

Xah Lee — Wed, 08 Feb 2012 16:48:02 +0000

has anyone studied the effect on math formulas if everything is expressed in just sine? e.g. like the new π, the τ site.

By: Howard

Howard — Sat, 12 Nov 2011 16:51:48 +0000

In right-triangle trigonometry, there is a combinatoric answer as well. How many ratios of two distinct sides are there when you have 3 sides? 3 x 2 = 6 trigonometric functions. In the medieval Islamic world they used all six and had very accurate tables of them; the sine, which is the earliest explicitly recognized, comes from India, where in the 1400s the infinite series expansion was known and given by Nilakantha, among others.

By: Thony C

Thony C — Wed, 09 Nov 2011 19:55:55 +0000

Historically the first trig function was the cord which the Indians replaced with the half cord or sine. Interesting in this context is the fact that the tangent is just as old but was known as the shadow function and was not considered to be a trig function until the middle ages.

By: College math in a single symbol — The Endeavour

College math in a single symbol — The Endeavour — Tue, 14 Dec 2010 13:00:07 +0000

[...] “the” hypergeometric function.) The most commonly used functions in application — trig functions, exp, log, the error function, Bessel functions, etc. — are either hypergeometric functions [...]

By: Xah Lee

Xah Lee — Wed, 27 Oct 2010 16:24:33 +0000

nice article. I thought about this too.

my fav answer is 1 too, from this perspective: they are all mathematically circular functions, captured by Sine. (Cosine is just the same function with a shift)

also enjoyed your explanation why historically more are important. Scan’d about them on wikipedia before but didn’t know the reason why.

By: CogitoErgoCogitoSum

CogitoErgoCogitoSum — Fri, 16 Apr 2010 09:40:33 +0000

I thought it noteworthy to mention that – at least for sine, cosine, tangent, cosecant, secant, and cotangent – we also have the hyperbolic counterparts. Thats an additional (at least) six.

Then looking back into history, there is also the chord function – a deprecated and all but forgotten trig function which can easily be expressed in terms of sines.

Its been a while since I looked, but I think the Wikipedia article on the trig functions lists probably one or two more, at least, than any listed thus far.

I liked Josh Reich’s answer… e and i. Indeed. I appreciate that answer. And yet, the relationships youre referring to could not have ever been proven without first defining the more fundamental trig functions… they would have been a natural chronological predecessor in the conceptual development of mathematics. Its highly, highly unlikely we would have derived the mathematics in the other direction.

If you get down to the real truth of the matter, there are no trig functions at all. There are just right triangles. Sides are proportional (which is what makes trig work in the first place) due solely to the fact that congruent angles makes for similar shapes.

By: John Armstrong

John Armstrong — Fri, 05 Feb 2010 19:52:19 +0000

2) Sine and Cosine may be related as you indicate, but they’re a very important pair. Specifically, they provide an orthogonal basis for the space of solutions to the differential equation

$latex \displaystyle\frac{d^2}{dt^2}u(t)+u(t)=0$

By: Wing

Wing — Tue, 29 Sep 2009 01:19:31 +0000

Thanks for the article. I just read the Wikipedia articles on versine, haversine, etc. and it’s pretty awesome. It’s sad that there’s really no way to actually learn spherical trig anymore in school.

As to why a calc text claims that there are 6 trig functions: a lot of formulas can be expressed more nicely with sec, cos and cot. The derivative of tan is for example secant squared. Also, the 1 + tan^2 =sec^2 identity plays a crucial role in some integrals.

I guess you have as many trig functions as you need. In most cases, it would probably be 0.

By: Sue VanHattum

Sue VanHattum — Sat, 26 Sep 2009 23:15:06 +0000

Thank you! One of the texts I’ve used said something about the most common 6 trig functions, and I was bewildered. I had never gotten around to researching this, so I’m delighted to have it drop into my lap today. Now I know. 6 more antiques to think on, if you wish.

By: tom

tom — Sat, 26 Sep 2009 02:50:56 +0000

I can see the argument for there being no trig functions – and revert back to e and i, but of course these are only elements in a field, not functions in of themselves. The ‘fundamental’ function relating e, i, and sines and cosines is of course the exponential function. For this matter, we can actually just return to sums and products and toss in the weird operation of ‘infinite’ sums of products; or perhaps the ‘limit’ of an infinite sequence of partial sums is even more basic.

By: Mark Durst

Mark Durst — Fri, 25 Sep 2009 21:48:27 +0000

Am now trying to think whether the co- in cosine can be expressed as a special case of the co- in everything else (cohomology, cobordism, etc.)

I’m told that at Mathcamp last year, two individuals agreed to share the job of coordinator. So they were, of course, called the ordinators.

By: josh reich

josh reich — Fri, 25 Sep 2009 19:07:24 +0000

There are no trig functions, only e and i.