I worked for a few years as life actuary. They study syllabus was a bit different then vs. as I vaguely understand it now. In that era, one studied math courses to become an associate of the Society of Actuaries. Then you studied the actual machinations of the insurance business to become a Fellow.
Passing exams was a key factor in your pay scale which was a plus or a minus depending on results. But for sure, one's career track is heavily dependent on these results.
But, in short, I can tell you that I never used one iota of the math covered on the exams. They are more of a training ground to come up with fast ways to solve problems. The higher level exams require a good deal or practice and are high-pressure considering what is at stake. As an example of the intensity, I got a 10 (out of 10) on the risk theory exam (considered a tough one) answering 7 out of 15 questions - and I yet may have gotten some wrong.
The major function of actuaries was to calculate and substantiate the adequacy of reserves. This relied heavily on mortality tables - an easy math problem. A significant aspect of the endeavor entailed accounting regulations both for GAAP and state of domicile statutory requirements.
Most of the actual work in this regard was modeling of various policy lines. And each company has it's own set of well- worn models.
A latter consideration was the requirement of cash flow testing. This again was a modeling problem as the investment assets had to be tested for their performance under various, usually extreme, interest rate scenarios. This was intended to make sure there is money available to fund the reserves.
So while passing exams entailed cubic splines, the Poisson distribution, the Black-Scholes model, Ito's Lemma, the real work required familiarity with insurance regulations and their related accounting treatment.
For more information on the study materials, esp. books, you can go to the websites of the Society of Actuaries and the Casualty Actuarial Society.
I've got three, which I've arranged in increasing order of broad-reaching real-world significance.
One is pedagogical, by which I mean it doesn't really have much to do with trigonometry itself but is rather an artifact of the order in which mathematics tends to be taught. This is a motivation for the rigor of calculus. After all, up to this point in a person's education, the only functions $\mathbb{R}\to\mathbb{R}$ that they are likely to have discussed as such are polynomials, exponentials, logarithms, and trigonometric functions. It is possible using purely classical means to solve derivatives and integrals of low-degree polynomials. Therefore, trigonometric functions are a key insight into understanding the richness of function space, and therefore a motivation for wanting to investegate it in more than an ad-hoc way.
The second is more exciting, which is that trigonometry gives us a basic insight into the nature of the complex numbers. You may know that it is reasonable to append to the reals a special number $i$ which has the property that $i^2=-1$. The number is not a real number, but it turns out that all of the arithmetic operations that are possible on reals are also possible on numbers of the form $a+bi$. But notice what that form looks like. Just as the properties of real numbers invite us to plot them as points on a line, this understanding of the complex numbers invites us to plot them as points in the plane. Once we do this, a new thought (eventually) occurs to us: these complex numbers are vectors. But vectors have magnitude and direction. If one works through the technical difficulties, one finds* the beautiful relationship
$$\text{Mag}(z)e^{i\text{Ang}(z)} = \text{Mag}(z)\sin(\text{Ang}(z))+i\text{Mag}(z)\cos(\text{Ang}(z)).$$
The right hand side of that equation is of the form $a+bi$, which means we have discovered the relationship between the so called "polar" form of complex numbers and the way in which they are traditionally defined. But you might ask why we should care about complex numbers, well,
The third is more sobering, and it is what I would argue is the most widely manipulated phenomenon that higher mathematics is interested in (and in fact still maintains an interest in!): the concept of a Fourier series. The story here is rich and complex, and I do not know all of it and if you would like to learn, the rest of the internet has much better words to say than I do. But I will state the basic idea: any sufficiently nice function can be arbitrarily well-approximated by a function of the form:
$$f(x)\approx \hat f(x)=\sum_{i=0}^N a_n\cos(nx)+b_n\sin(nx)$$
This is perhaps surprising, but if you fiddle with some explicit approximations then you could become convinced of this eventually. What this means physically is that a properly encoded device could read the various $a_n$ and $b_n$ and reproduce the function arbitrarily well. More to the point, it means that if you want to tell your function to someone else (which in general contains an infinite amount of information), they can get arbitrarily fine detail if you just send them a finite number of the coefficients. Therefore, one can send signals over long distances with relative ease, and this is the method that makes radio, telephone, and the internet work. So the reality is that little trigonometry, with humble origins on the unit circle, is now responsible for making our "information society" possible.
(Ah, I did promise a connection to complex numbers. But perhaps you see it already: just as sines/cosines can be written as complex numbers, the opposite is true as well. In fact, this ends up being the "right" way to think of Fourier series mathematically, in that it admits incredibly, almost unbelievably, far-reaching generalizations. In the interest of keeping your attention, I have omitted these; they are quite technical and to fill you in on the background details would have more than tripled the size of this answer.)
(*In fact the story is slightly more complicated than this: the truth is that this is how one traditionally defines complex exponentials, but there are good reasons for doing so: things which one wants to be true about exponentials are true for this particular combination of sines and cosines, except for some technical obstructions that in any case no reasonable definition would be able to overcome.)
Best Answer
My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 probability to be rainy; if today is sunny, then tomorrow has a 0.5 probability to be rainy. Then you may organize these data into a matrix:
$$A=\begin{pmatrix} 0.9 & 0.5 \\ 0.1 & 0.5 \end{pmatrix}$$
Now if you compute $A^2=\begin{pmatrix} 0.86 & 0.7 \\ 0.14 & 0.3 \end{pmatrix}$, what do you get? 0.86 is the probability that if today is rainy then the day after tomorrow is still rainy and 0.7 is the probability that if today is sunny then the day after tomorrow is rainy. And this pattern holds for $A^n$ an arbitrary $n$.
That's the simple point: matrices are a way to calculate elegantly. In my understanding, this aligns with the spirit of mathematics. Math occurs when people try to solve practical problems. People find that if they make good definitions and use good notations, things will be a lot easier. Here comes math. And the matrix is such a good notation to make things easier.