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, there’s often a need to memorize many new names. Hence, this time we have prepared a menu related to names.

Trigonometric Functions

Trigonometric functions are one of the most encountered functions in science and engineering studies. However, there might not be many people who know the origin of their names. As far as I remember, I first learned about them in middle school, but since they were taught at such a young age, I might have accepted them as facts and missed 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 functions are fundamentally related to triangles, and their names are derived from their geometric meanings. Their inverse functions are sometimes denoted as $\sin^{-1}, \cos^{-1}$ , but the notation $\arcsin, \arccos$ is used more frequently. Why were such names given?

In the trigonometric function family, there are also functions known as hyperbolic functions. Their notations and definitions are as follows.

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

Detailed information related to these can be found below.

Geometry

In mathematics, there are many names attached to the term “geometry.” Starting with geometry, there are Euclidean geometry, differential geometry, geometric mean, geometric series, and geometric distribution. However, if you look closely, those that aren’t “geometry” don’t seem closely related to geometry. The ones that don’t intuitively appear related to geometry are as follows.

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

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

Multiplication signifies an area, which connects it to geometry.
If the product of two different numbers equals the square of one number, then that number is the (multiplicative) mean.

For more details, refer to the documents below.

A geometric series is the infinite sum of a geometric sequence with the first term $a$ and common ratio $r$ . It is referred to as a geometric series because the $n$ th term is the geometric mean of the $n-1$ th and $n+1$ th terms.

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

A geometric distribution is called such because its probability mass function takes the form of a geometric sequence.

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

It is the $x$ th term of a geometric sequence with the initial term $p$ and ratio $1-p$ , and because geometric sequences are related to the geometric mean as seen above, this distribution is called geometric distribution. Thus, the word geometric naturally attaches to concepts related to multiplication. Particularly, the phrase “increasing geometrically” is often used in everyday speech to mean increasing at a very rapid rate. In the graph below, you can see the difference between increase by addition ( $2+2+2+2+\cdots$ ) and increase by multiplication ( $(2 \times 2 \times 2 \times 2 \cdots)$ ).

Miscellaneous

We have prepared other menus that are difficult to categorize into one group. These items include those related to notation.