How many trig functions are there?

How many basic trigonometric functions are there? I will present the arguments for 1, 3, 6, and at least 12.

The calculator answer: 3

A typical calculator has three trig functions if it has any: sine, cosine, and tangent. The other three that you may see — cosecant, secant, and cotangent — are the reciprocals of sine, cosine, and tangent respectively. Calculator designers expect you to push the cosine key followed by the reciprocal key if you want a secant, for example.

The calculus textbook answer: 6

The most popular answer to the number of basic trig functions may be six. Unlike calculator designers, calculus textbook authors find the cosecant, secant, and cotangent functions sufficiently useful to justify their inclusion as first-class trig functions.

The historical answer: At least 12

There are at least six more trigonometric functions that at one time were considered worth naming. These are versine, haversine, coversine, hacoversine, exsecant, and excosecant. All of these can be expressed simply in terms of more familiar trig functions. For example, versine(θ) = 2 sin2(θ/2) = 1 – cos(θ) and exsecant(θ) = sec(θ) – 1.

Why so many functions? One of the primary applications of trigonometry historically was navigation, and certain commonly used navigational formulas are stated most simply in terms of these archaic function names. For example, the law of haversines. Modern readers might ask why not just simplify everything down to sines and cosines. But when you’re calculating by hand using tables, every named function takes appreciable effort to evaluate. If a table simply combines two common operations into one function, it may be worthwhile.

These function names have a simple pattern. The “ha-” prefix means “half,” just as in “ha’penny.” The “ex-” prefix means “subtract 1.” The “co-” prefix means what it always means. (More on that below.) The “ver-” prefix means 1 minus the co-function.

Pointless exercise: How many distinct functions could you come up with using every combination of prefixes? The order of prefixes might matter in some cases but not in others.

The minimalist answer: 1

The opposite of the historical answer would be the minimalist answer. We don’t need secants, cosecants, and cotangents because they’re just reciprocals of sines, cosines, and tangents. And we don’t even need tangent because tan(θ) = sin(θ)/cos(θ). So we’re down to sine and cosine, but then we don’t really need cosine because cos(θ) = sin(π/2 – θ).

Not many people remember that the “co” in cosine means “complement.” The cosine of an angle θ is the sine of the complementary angle π/2 – θ. The same relationship holds for secant and cosecant, tangent and cotangent, and even versine and coversine.

By the way, understanding this complementary relationship makes calculus rules easier to remember. Let foo(θ) be a function whose derivative is bar(θ). Then the chain rule says that the derivative of foo(π/2 – θ) is -bar(π/2 – θ). In other words, if the derivative of foo is bar, the derivative of cofoo is negative cobar. Substitute your favorite trig function for “foo.” Note also that the “co-” function of a “co-” function is the original function. For example, co-cosine is sine.

The consultant answer: It depends

The number of trig functions you want to name depends on your application. From a theoretical view point, there’s only one trig function: all trig functions are simple variations on sine. But from a practical view point, it’s worthwhile to create names like tan(θ) for the function sin(θ)/sin(π/2 – θ). And if you’re a navigator crossing an ocean with books of trig tables and no calculator, it’s worthwhile working with haversines etc.

Related posts:

Mercator projection
Why care about spherical trig?
Three trigonometry topics
What is the cosine of a matrix?
Connecting trig and hyperbolic functions without complex numbers

17 thoughts on “How many trig functions are there?

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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

    $latex displaystylefrac{d^2}{dt^2}u(t)+u(t)=0$

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. Interesting article. First time I ever heard of those archaic “ha-” and “ver-” functions.

    In my opinion there are 24 “practical” trigonometric functions, 12 being classic trig functions and 12 being hyperbolic trig functions.

    – 6 basic trig functions: sin, sec, tan, cos, csc, cot
    – 6 inverse functions of the above: arcsin, arcsec…
    – 6 hyperbolic trig functions: sinh, sech, tanh…
    – 6 inverse functions: arcsinh, arcsech, arctanh…

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