What are some interesting and unusual theorems, identities, notations in Trigonometry?

Question

Context/Motivation

When studying a subject I usually like to create a document and make a kind of study guide, notes about the subject. I created the document and write using LaTeX. Also, besides learning about the subject itself (it is math or something related to robotics) I practice LaTeX and English because I write it in English. Taking trigonometry, the topic of this question, I would write lecturing myself and go through concepts, theorems, formulas detailing as much as possible. Yesterday night I was wondering about making notes about trigonometry and it reminded me of a notation that I did find unusual when I first saw it, but it was interesting regarding the Domain of the tangent function.

Working with trigonometry function such that $f:\mathbb{R}\rightarrow\mathbb{R}$, we have $f(x)=\tan(x)=\dfrac{\sin(x)}{\cos(x)}$. As $\cos(x)\neq 0$ the domain of the tangent function is usually written as $\operatorname{Dom}(\tan) = \left\{x: x \neq \dfrac{\pi}{2}+k\pi, k \in \mathbb{Z} \right\}$.

We have $\cos(x) = 0$ for $x=\pm\dfrac{\pi}{2}, \pm\dfrac{3\pi}{2}, \pm\dfrac{5\pi}{2}...$

In fact, taking the intervals we have $$... \quad \cup\left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right)\cup \quad ...$$

For $k\in\mathbb{Z}$. Considering $x_k = \dfrac{\pi}{2}+k\pi = \dfrac{2k\pi+\pi}{2}= \boxed{\dfrac{(2k+1)\pi}{2}}$, we can note that the intervals are determined by these $x_k$ points, so

$$(x_k, x_{k+1})= \left(\dfrac{(2k+1)\pi}{2}, \dfrac{(2k+3)\pi}{2}\right)$$

Once we have an infinity quantity of unions, we can represent it using the big union notation. This is a notation I was not familiar, that is why I found it very interesting. Therefore, we have

$$\operatorname{Dom}(\tan) = \left\{x: x \neq \dfrac{\pi}{2}+k\pi, k \in \mathbb{Z} \right\} = \bigcup_{k=-\infty}^{\infty} \left(\dfrac{(2k+1)\pi}{2}, \dfrac{(2k+3)\pi}{2}\right)$$

Question

I'm afraid this question might be off-topic but I am asking this here because it seems to be a topic where experience is very important. I am looking identities, theorems, concepts, notations that are not very common, usual in regular books, textbooks. It is just out of my curiosity, but I am really interested in these new things. I'm sure the example I gave above might not be impressive for university students or most people who use MSE, but I was very surprised when I first saw it.

More precisely, I am looking for answers of the following kind:

Trigonometric Identities that are unusual that might/might not have interesting applications;

Notations that are not very common;

Concepts, Theorems that are usually not seem is regular high-school but actually presents interesting ideas;

University trigonometry concepts with interesting ideas/application in high schools trigonometry;

The question is certainly very wide regarding the topic, but any interesting information is very welcomed.

Answer 1

If you are interested in some interesting and significant trigonometric identities, then my file Special Algebraic Identities (ident04.gp) may have what you are looking for. The file has over $600$ identities some of which have specific tags.

For example, the tag [TS] indicates

The functional equation has trigonometric sine function solutions. Note that this includes circular and hyperbolic sine functions.

The tag [TT] indicates

The functional equation has trigonometric tangent function solutions. Note that this includes circular and hyperbolic tangent functions.

There are six such tags which also specialize to trigonometric identites.

A example of such an identity with tag [JE] is id3_4_1_3a which is $$ a b (a - b) + b c (b - c) - a c (a - c) - (a - b) (a - c) (b - c) = 0 $$ The functional equation associated with this identity is $$ f(a)f(b)f(a-b)+f(b)f(c)f(b-c)-f(a)f(c)f(a-c)-f(a-b)f(a-c)f(b-c) = 0. $$ This is satisfied by $\,f(x) = \sin(x)\,$ and also $\,f(x) = \tan(x).$ These are special cases of the identity for $\,f(x) = \text{sn}(x,m), \,$ the Jacobi elliptic function sn.

Answer 2

Here are a couple of websites that may be of interest to you...

The Forgotten Trigonometric Functions dealing with trigonometric functions used for navigations, such as the versine, the coversine, the haversine, and others.

Real Life Applications of Trigonometry

Answer 3

Trigonometry is based on relationships between the side lengths $a,b,c$ and corresponding angles $\alpha,\beta,\gamma$ of triangle, but sometimes it is more convenient to use express these relations in terms of another three linear properties of triangle, namely semiperimeter $\rho=\tfrac12(a+b+c)$, inradius $r$ and circumradius $R$ of the triangle.

It is especially useful, that trigonometric expressions for the angles can be formulated in terms of just two unitless parameters, $u=\tfrac\rho{R}$ and $v=\tfrac r{R}$, which can be considered as the semiperimeter and the inradius of triangle, inscribed in the unit circle ($R=1$). This is useful since the ranges of these two parameters $u,v$, that correspond to a valid triangle (including degenerate cases), are bounded as

\begin{align} v&\in[0,\tfrac12] ,\quad u(v)\in[u_{\min}(v),u_{\max}(v)] \tag{1}\label{1} ,\\ u_{\min}(v)&=\sqrt{27-(5-v)^2-2\sqrt{(1-2v)^3}} \tag{2}\label{2} ,\\ u_{\max}(v)&=\sqrt{27-(5-v)^2+2\sqrt{(1-2v)^3}} \tag{3}\label{3} . \end{align}

Curves \eqref{2}, \eqref{3} and the line $v=0$ bound the region, for which any pair $(v,u)$ correspond to unique triangle, that is a triple of angles $\alpha,\beta,\gamma$. Any pair $(v,u)$ outside that region can not define (be a properties of) a valid triangle. The boundary curves $u_{\min}(v)$, $u_{\max}(v)$ correspond to isosceles triangles, the line $u(v)=v+2$ inside the region represent right-angled triangles, the point where $u_{\min}(v)=u_{\max}(v)=u_{\min}(\tfrac12)=u_{\max}(\tfrac12)=\tfrac32\sqrt3$ corresponds to equilateral triangle.

Not surprisingly, any triplet of the trigonometric functions of $\alpha,\beta,\gamma$ can be found as a solution to the cubic equation \begin{align} x^3-a_2x^2+a_1x-a_0&=0 \tag{4}\label{4} , \end{align}

where $a_i$ can be expressed in terms of $u,v$.

For $\cos(\cdot)$ we have: \begin{align} a_0&=\cos\alpha\cos\beta\cos\gamma =\tfrac14(u^2-(v+2)^2) \tag{5}\label{5} ,\\ a_1&= \cos\alpha\cos\beta+\cos\beta\cos\gamma+\cos\alpha\cos\gamma =\tfrac14(u^2+v^2)-1 \tag{6}\label{6} \\ a_2&=\cos\alpha+\cos\beta+\cos\gamma =v+1 \tag{7}\label{7} . \end{align}

For $\cot(\cdot)$ we have: \begin{align} a_0&=\cot\alpha\cot\beta\cot\gamma =\frac{u^2-(v+2)^2}{2u\,v} \tag{8}\label{8} ,\\ a_1&=\cot\alpha\cot\beta+\cot\beta\cot\gamma+\cot\gamma\cot\alpha =1 \tag{9}\label{9} ,\\ a_2&=\cot\alpha+\cot\beta+\cot\gamma = \tfrac12\,\left(\frac{u}{v}-\frac{v}{u}\right)-\frac{2}{u} \tag{10}\label{10} , \end{align}

and so on.