Spring 2025 Omakase: Your Name 📂JOF

Spring 2025 Omakase: Your Name

Introduction

In Korea, the new academic year starts in March, so around this time, you have many new names to learn. Therefore, we have prepared a menu related to names today.

Trigonometric Functions

Trigonometric functions are some of the most encountered functions when studying STEM fields. However, not many may be well-versed with their names. As I recall, I first learned about them in middle school, but perhaps because we learn them at such a young age, we accept them as they are and miss the chance to wonder about their names.

The basic trigonometric functions are as follows:

$\begin{align*} \sin \theta &:= \dfrac{y}{\sqrt{x^{2}+y^{2}}} \\ \cos \theta &:= \dfrac{x}{\sqrt{x^{2}+y^{2}}} \\ \tan \theta &= \dfrac{y}{x} = \dfrac{\sin \theta}{\cos \theta} \end{align*}\qquad\qquad \begin{align*} \sec \theta &:= \dfrac{1}{\cos \theta} = \dfrac{\sqrt{x^{2}+y^{2}}}{x} \\ \csc \theta &:= \dfrac{1}{\sin \theta} = \dfrac{\sqrt{x^{2}+y^{2}}}{y} \\ \cot \theta &:= \dfrac{1}{\tan \theta} = \dfrac{x}{y} = \dfrac{\cos \theta}{\sin \theta} \end{align*}$

These are fundamentally related to triangles and have their names derived from geometrical meanings. Their inverse functions are sometimes denoted as $\sin^{-1}, \cos^{-1}$ , but the notation $\arcsin, \arccos$ is more widely used. Why are these names given, though?

Among the trigonometric function family, there are also functions called hyperbolic functions. Their notation and definition are as follows:

$\begin{align*} \sinh \theta &:= \dfrac{e^{\theta} - e^{-\theta}}{2} \\ \cosh \theta &:= \dfrac{e^{\theta} + e^{-\theta}}{2} \\ \tanh \theta &:= \dfrac{\sinh \theta}{\cosh \theta} = \dfrac{e^{\theta} - e^{-\theta}}{e^{\theta} + e^{-\theta}} \\ \end{align*}$

You can find more detailed content related to these below.

Geometry

In mathematics, there are many terms with the word “geometry” attached. From geometry itself to Euclidean geometry, differential geometry, geometric mean, geometric series, and geometric distribution. Upon closer examination, items not labeled as “geometry” do not seem to relate closely to geometry. Intuitively, the things that seem unrelated to geometry are as follows:

$\begin{align*} \footnotesize \text{기하 평균: }& \sqrt{ab} \\ \footnotesize \text{기하 급수: }& \sum_{n=0}^{\infty} ar^{n} = a + ar + ar^{2} + ar^{3} + \cdots \\ \footnotesize \text{기하 분포: }& p(x) = p(1-p)^{x-1}, \qquad (x = 1, 2, 3, \dots) \\ \end{align*}$

A simple explanation of why $\sqrt{ab}$ is called the geometric mean is as follows:

Multiplication signifies area, which essentially represents geometry.
If multiplying two different numbers is equal to squaring one of them, that’s the (multiplicative) mean.

For further details, please refer to the document below.

A geometric series is the infinite sum of a geometric sequence $\left\{ ar^{n-1} \right\}$ with a first term of $a$ and a common ratio of $r$ . It is known as the geometric series because the $n$ th term is the geometric mean of the $n-1$ th term and the $n+1$ th term.

$ar^{n} = \sqrt{(ar^{n-1})(ar^{n+1})}$

The reason why the geometric distribution is called a geometric distribution is due to its probability mass function having the form of a geometric sequence.

$p(x) = p(1-p)^{x-1}, \qquad x = 1, 2, 3, \dots$

The sequence is the $x$ th term of a geometric sequence with a first term of $p$ and a common ratio of $1-p$ . As we saw above, geometric sequences relate to geometric means, hence this is called a geometric distribution. This is why concepts related to multiplication naturally have the word geometric attached. Particularly, the phrase “grows geometrically” is often used in everyday language to imply an extraordinarily rapid increase. The following illustration effectively shows the difference between increase by addition ( $2+2+2+2+\cdots$ ) and increase by multiplication $(2 \times 2 \times 2 \times 2 \cdots)$ .

$\cdot | \cdot$ Notation

In mathematics, notations like $A | B$ are primarily used when considering a target, $A$ , with restrictions that meet a certain condition, $B$ . It is especially familiar from high school days. For example, the probability that a certain event $A$ occurs is expressed as $P(A)$ , and the conditional probability of $A$ occurring given that event $B$ already occurred is expressed as $P(A | B)$ . A similar context is used in set notation with conditions. A set of vectors in $n$ dimensions, all having a magnitude of $1$ , is expressed as follows:

$\left\{ \mathbf{x} \in \mathbb{R}^{n} | \left\| \mathbf{x} \right\| = 1 \right\}$

Here, the left side of $|$ signifies the target, and the right side indicates the condition. In the case of sets, instead of using the bar ( $|$ ), notation using a colon ( $:$ ) is also widely used. Another example can be found in restrictions. When there is a function $f : X \to Y$ , a function defined with the domain restricted to $A$ is denoted as $f|_{A}$ . That is, $f|_{A}$ shares the same functional value as $f$ , but its domain condition is given as $A$ .

Conditional Probability $P(A | B)$
Set Notation with Conditions $\left\{ \text{elements} | \text{conditions} \right\}$
Restrictions $f|_{A}$

As shown in these examples, the position—whether to the left or right of $|$ —plays a crucial role in the meaning of $A|B$ notation. Although not in the same context as previously explained, in number theory, when $a$ divides $b$ , it is expressed as $a|b$ using notation, which significantly differs in meaning from $b|a$ .

Miscellaneous

We have prepared some items here that are challenging to categorize into one specific category. They include notations or names that might not quite align with intuition.