# Why do we need so many trigonometric definitions?

Examples $$\sec(x) = \dfrac{1}{\cos(x)}$$ $$\cot(x) = \dfrac{1}{\tan(x)}$$

There are many more out there, but why do we need definitions that can be written with just $\sin , \cos ,\tan$ etc. in maths? Why can't they just be written as their expanded form?

Most trigonometric functions can be written with just $\sin \cos$ and $\tan$. Why do we need so many?

• I'm not sure, but I think these are left over from an earlier time when trigonometry was practiced more geometrically and with less of an algebraic emphasis. The name of the "secant" is another vestige of this time: a "secant" is a line that cuts a curve at two points. What line does the "secant" function refer to? ( I don't know.) – MJD Mar 29 '18 at 16:28
• While the title is misleading, the question is valid: we (at least seem) to have an unnecessary surplus of trigonometric functions. We could get away with just $\sin$ and $\cos$ for most purposes if we wanted. Which begs the question why they were chosen to be named. This question definitely ran through my mind when first learning trigonometry. I disagree with the downvotes, close votes, and find the top three comments here to be unhelpful. – anon Mar 29 '18 at 16:30
• Let's take it further, @PrzemysławScherwentke: why do you need $cos(x)$? It is $sin(x+\frac{\pi}{2})$ – Ottavio Bartenor Mar 29 '18 at 16:32
• Indeed, someone did ask why we need both sine and cosine, to the tune of a 116 question score, 16k views, and being the #3 question in the (trigonometry) tag. – anon Mar 29 '18 at 16:44
• Your title should explicitly describe the content of the question. "Why do we need definitions?" is a bad title since it a) doesn't describe what you're actually asking and b) is actively misleading about what the content is. Voting to close as unclear now. – user296602 Mar 29 '18 at 16:55

There are a lot of trigonometric functions which are defined geometrically, which we rarely use anymore. Many of these are summarized by this image:

These all have their uses in particular circumstances. For example, the half versed sine (or haversine) is useful for determining the great circle distance between points, which is incredibly useful if you are trying to navigate. We don't need the haversin, but it is useful, and reduces notation a bit in at least one specific context. The other trig functions are similar—personally, I would rather write $$\frac{\mathrm{d}}{\mathrm{d}t} \tan(t) = \sec(t)^2$$ than $$\frac{\mathrm{d}}{\mathrm{d}t} \frac{\sin(t)}{\cos(t)} = \frac{1}{\cos(t)^2}.$$

EDIT: This answer was written when the question seemed to be asking about the "necessity" defining secant and cotangent functions. It seems that the original questioner had a much more general question in mind, i.e. why do we need any definitions at all? The only possible response that that, I think, is because mathematics would be impossible without "definitions." Working under the assumption that the original questioner is in earnest, a partial answer is as follows:

A huge part of mathematics is the language we use in order to communicate mathematical ideas. We could, I suppose, never define anything beyond the basic axioms, but then we could never get anything done, and would have no hope of ever communicating our ideas to others. If we don't define a derivative, how do we describe the the motion of a planet? It would be cripplingly inconvenient if we could never write $3$, and always had to write $\{ \{\}, \{\{\}, \{\{\}\} \}, \{\{\}, \{\{\}, \{\{\}\}\} \}$. Not only is that quite hard to read (do you really want to check that I got all of my commas and braces right?), it is horribly inefficient. And this is just to describe a relatively small natural number. It only gets worse from here!

The point is that definitions allow us to encapsulate complicated ideas into a short collection of symbols (i.e. words) that allow us to make further deductions. Definitions are at the very heart of mathematics. We can do nothing without them.

• @RosieF I don't find $\sec^2 t$ to be any simpler than $\sec(t)^2$, and the former has the problem of being potentially ambiguous: I work with iterated function systems where $f^n = f\circ f \circ \cdots f$, i.e. the $n$-fold composition of $f$ with itself. – Xander Henderson Mar 29 '18 at 20:46
• One good reason one had such function (for use in navigation and practical astronomy) is that they were tabled. Nice to have a name for your table, And, in a time where even simple arithmetic where complicated (especially on a sailing vessel in full storm), one wanted tables for the end result you needed! – kjetil b halvorsen Mar 29 '18 at 21:19
• About the derivative of $\tan$, I actually prefer $1+\tan^2$. It shows that $\tan$ solves the differential equation $y'=1+y^2$ and easily gives us $\arctan'(x)=\frac{1}{1+x^2}$. – Torsten Schoeneberg Mar 29 '18 at 23:18
• @MilesRout Actually, I think that your comment further emphasizes the point. Yes, "3" is an abstract concept, but so is everything in mathematics. We need definitions in order to pin down the abstract ideas so that we can actually do stuff with them. Of course, depending on how one builds the natural numbers, there are other ways of defining the object 3 (say, $S(2)$, where $S$ is the successor (what's that? oh, no! more definitions!), $2 = S(1)$, 1 = S(0)$, and$0$is defined axiomatically. Or$(((0+1)+1)+1)$(where$0$,$1$, and$+$all need to be defined). Or... – Xander Henderson Mar 31 '18 at 13:06 • @MilesRout I respectfully disagree. For my purposes, the nested sets make my point better (in my opinion). I'm not trying to teach a course in foundations, I just need a quick example that emphasizes how definitions can be used to wrap up complicated ideas in a little package to make life easier. Whether we write 3 as a set or as the successor of something, the point is the same. – Xander Henderson Mar 31 '18 at 13:17 You could equally ask why we define "$8$" when we could just write$1+1+1+1+1+1+1+1$. It's convenient to have shorter names for things that get used a lot. Admittedly, "$8$" is much more convenient (and much more often used) than something like$\sec$, so there's always room to argue about whether a particular abbreviation is really useful. • That strongly depends on what you're writing. My undergraduate maths course probably had fewer total uses of$8$in it than uses of$\tan$. Of course$2\pi$turned up quite frequently, which leads to another disputed definition... – origimbo Mar 29 '18 at 20:36 • @origimbo Obviously, the dispute you refer to is whether or not one should write$2\pi$,$(1+1)\pi$or even just$\pi+\pi$. ;-) – David Richerby Mar 29 '18 at 20:54 The definitions have existed for a long time and basically the reason we write$\tan(x) =\frac{\sin(x)}{\cos(x)}$or$\sec(x) = \frac{1}{\cos(x)}$etc. is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say$\sec(x)$values than calculate$\frac{1}{\cos(x)}$in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family. I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown (Tattoos On Math) and the other is from an amazing guy called Simon Clark. (Why$\sin$and$\cos$don't mean anything). Edit: Forgot to mention, to be honest$\sin(x)$and$\cos(x)$are the only trigonometric values we need, the rest can be derived. But the world is sometimes a real scary place without$\tan(x)$,$\cot(x)$,$\sec(x)$and$\operatorname{cosec}(x)$. • What a title : Why sin and cos don't mean anything ... – Peter Mar 29 '18 at 16:37 • Yep XD .Thats why i like him,his videos are fun. – The Integrator Mar 29 '18 at 16:41 • Well, apparently tastes are different. Such a title would prevent me to watch the video. – Peter Mar 29 '18 at 17:02 • And indeed we don't even need$\cos$because we could write$\sin (\pi/2-x)$or (provided that signs are dealt with correctly)$\sqrt{1-\sin^2 x}$for$\cos x$. – Rosie F Mar 29 '18 at 20:36 • To be honest,$\sin$and$\cos$are completely useless. Everything should be defined in terms of$\tan$and$\operatorname{exsin}$. – Xander Henderson Mar 31 '18 at 21:33 In terms of their range of values on $$[-\infty,\,\infty]$$, $$\sin$$ is analogous to $$\tanh$$, $$\cos$$ to $$\mathrm{sech}$$ etc. These relations are formalized with the Gudermannian function, which notably connects circular trigonometric functions to hyperbolic ones without complex numbers. Having "unnecessary" function names not only makes these relations neater, it also gives functions partners with the same range, rather than comparing one function to the reciprocal of another. • I had not seen the Gudermannian function before. I am happy to have been introduced to it. Thank you. – Xander Henderson Mar 29 '18 at 16:46 • @XanderHenderson You're welcome. I introduced it to my PhD supervisor when I used it in Sec 1.8.1 of my thesis, which you can see here: etheses.whiterose.ac.uk/12277/1/… – J.G. Mar 29 '18 at 16:55 • @J.G. What is the integral of gd? Maple gives me complex stuff but it should be real-valued when$x$is real... – max_zorn Mar 31 '18 at 20:45 • @max_zorn Maple's giving you a$z+z^\ast$expression, which is real. – J.G. Mar 31 '18 at 21:11 • @J.G. I would like one without any imaginary stuff in it. I realize this is real. – max_zorn Mar 31 '18 at 21:15 It is indeed only a definition to handle trigonometric expressions and functions in other forms or to give a particular geometrical meaning to some expression as for example$\tan x = \frac{\sin x}{\cos x}$. The basic and foundamental trigonometric functions are$\sin x$and$\cos x$and the others are derivated from these. • If we only have$\sin(x)$, we cannot determine$\cos(x)$in general. But in suitable ranges, this is possible. – Peter Mar 29 '18 at 16:29 • @Peter Yes of course! we need at least 2 information! I've fixed that, Thanks – gimusi Mar 29 '18 at 16:30 • @Peter indeed I was thinking to the case in which$x$is given then$\sin x$is given and$\cos x = \pm \sqrt{1-\sin^2 x}$with the sign depending on$x$. – gimusi Mar 29 '18 at 16:36 They are not strictly necessary. It is just a matter of convenience. It is sometimes convenient to use$\sec(x)$instead of$1/\cos(x)$. You can live without them, but sometimes life is easier with them. The full list of trigonometric functions is much longer: versine, coversine, haversine, havercosine, and the inverse and hyperbolic functions. Some of the have fallen completely out of use, some partially. The secant, cosecant, and cotangent are a bit of an edge case: not very common, but they still appear occasionally. The tangent comes back (wearing a different hat) in calculus. However, you need these definitions. Let's pick one example. Suppose we only know$\cot \theta = \frac{1}{\tan \theta}$. Then cotangent is undefined whenever tangent is undefined or zero, which is every integer multiple of$\pi/2$. But if we know$\cot \theta = \frac{\cos \theta}{\sin \theta}$, we find out cotangent is only undefined when sine is zero, which is every even integer multiple of$\pi/2$. This is practice for a pattern that happens over and over -- different definitions of a function agree where they are both defined, but may each be defined in places where the others are not. In fact, there is a reasonable chance you have already done this twice with sine and cosine (and not really noticed because it might not have been pointed out). Originally, you only had trig functions defined for acute angles in geometric triangles -- only defined for angles$\theta$with$0 < \theta < \pi/2$. Then you extended these to be defined on the unit circle for$0 \leq \theta \leq 2\pi$by using the$\sin \theta = y, \cos \theta = x, \tan \theta = y/x, \dots$definitions. Then you extend these to all real angles$-\infty < \theta < \infty\$ by observing coterminal angles all intersect the unit circle at the same point. That's three definitions -- one using geometric triangles, one using coordinates on a unit circle, and one using coterminal angles to extend from a definition with a small domain to a larger domain.

This is a common activity. Limits of indeterminate forms are attempts to extend the domain of the difference quotient to the derivative and to extend the domain of (finite) Riemann sums to the integral. Analytic continuation is more of this. It goes on and on.

The fundamental lesson is that there is some sort of Platonic ideal abstract function, but each recipe we can find to evaluate it only tells us about the values on some subset of its domain. Different definitions cover different domains. To really be able to work with a function you need to be able to go every it can go -- not just everywhere one recipe for it can go.

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