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It seems that the following functions are not only excluded from a course in trigonometry, they are almost never taught in any course:

1. Chord


2. Versine


3. Coversine


4. Haversine


5. Exsecant


6. Excosecant

I could have asked the same question with the title "why have these functions lost their popularity" at math.stackexchange, but I fear that they will consider this question as "opinion-based".

More of a comment than an answer but: They are all composites of more basic functions. In fact, all of the trig functions could can expressed in terms of sine, linear changes in coordinates, and rational functions. For instance:

$$
tan(theta) = frac{sin(theta)}{sin( frac{pi}{2}-theta)}
$$

We certainly don't want children to have to memorize 20 different trig functions. That seems a bit silly. Do we even really need to teach all six ``standard'' trig function? Personally I tend to avoid $csc$ and $cot$ in my work...

As several respondents have indicated, one could choose many different trigonometric functions to serve as the basic elements in terms of which other trigonometric functions are expressed and in (ancient) history other choices were made than those standard now. What we know now that was not known to the ancients that leads us to use cosine and sine is the central place in mathematics of linear ordinary differential equations and exponential functions.

The functions $cos{t}$ and $sin{t}$ are a basis for the space of solutions of $ddot{x} + x = 0$, which is probably the most important differential equation in mathematics, because it is probably the most fundamental differential equation in physics. An alternative basis is $e^{i t}$ and $e^{-i t}$, so, said another way, the choice of $cos$ and $sin$ is made because these are the real and imaginary parts of the complex exponential $e^{i t} = cos{t} + i sin {t}$. The exponential is clearly fundamental, and it is convenient to express as much as one can in terms of it.

The alternative trigonometric functions do not have such properties and this means that to make extensive use of them it turns out to be more convenient to express them in terms of cosine and sine, or, equivalently, exponentials.

Also, this point of view makes apparent the parallelism with real exponentials, or rather the hyperbolic cosine and sine, corresponding to the differential equation $ddot{x} - x = 0$.

Moreover $cos$ and $sin$ admit simply geometric interpretations and sets of functions such as ${cos{nt}, sin{nt}: n in mathbb{Z}, n > 0}$ exhibit useful orthogonality properties (with respect to integration) that make them well adapted for use in contexts like Fourier series. There are of course many systems of orthogonal functions. What the most useful ones have in common is an origin as the solution of a homogeneous second order linear ODEs with some parameter (in this case $n$) in its coefficients.

What one teaches to children is in part dictated by what is needed later by those who will study more, and in part dictated by what has proved most tractable in diverse contexts.

Ask yourself if you would miss anything useful if you didn't know the functions you mentioned. I doubt it! I'm teaching high school math happily without having heard of them up to this point.

To the contrary, there are several downsides to teaching them:

1. 
   Quoting Steven Gubkin's comment: "Note that teaching students haversine, versine, etc will actually make it harder for them to communicate with others, because most other people have not learned these things."


2. Cognitive overload. Obligatory xkcd.


3. Personal opinion: Math should be more about (creative) problem solving than about applying memorized formulas. By not teaching above functions, students can solve more problems by creative usage of $sin$, $cos$, $tan$ and Pythagorean Theorem.


4. Calculators. If I express a solution using one of the fancy functions, I'm stuck with a calculator and can't come up with an approximate solution.

They are almost never needed for applications problems in physics or engineering. Really sine, cosine and tangent are mostly what you need. Not even the reciprocal functions.

I'm talking with respect to how formulas and problems are normally written in those courses. Ask yourself, did you encounter those functions in any of your science course and see a need for math to cover them as a service?

I think the versine and such are useful in navigation. But even for celestial nav as of post ww2, it was mostly done with tables and worksheets that don't require you to use these functions (or really know any of the math in what you do). I believe there was a time when there was more need for them before celestial nav became so work sheet oriented. And now celestial nav itself is a dying art because GPS is so common. Ask the average QM to use a sextant and see how he does...