The history of three old friends. Sine, Cosine, and Tangent

History

There are many mathematical concepts that cause a headache not only for students but also for their parents. Every time when you hear the words sine, cosine, and tangent, you are trying to recall the sides of a square triangle and how they relate to each other in the formulas.

Somehow, these words cannot be mentioned without the name of Pythagoras. If you want to understand the meaning of these terms, you need to understand the Pythagorean theorem and square triangles

Before we give the definitions of sine, cosine, tangent and other trigonometric functions, lets just take a look at their history. Though the terms are related to each other and often defined through each other, they have been proposed by different mathematicians from different countries.

• The term "sine" comes from the Latin word "sinus" which is a translation of the Arabic word "jayb" meaning "fold" or "bay". But in Arabian language the original word came from India. Indian mathematicians were using the "jya" function which meant "half chord". So the chain of words "jya > jaib > sinus" is the origin of the term "sine". The roots of the word "sine" are traced back to India. Because the word came from India, we may say that the law of sine was discovered there. At least Indian mathematicians were among the first mathematicians who discovered the properties of the sine.
• "Cosine" term doesn't come from India or Middle East. It was proposed by English mathematician Edmund Gunter. In 1620 he decided to name the function as a complement of sine. Indeed, the cosine and sine sound similar. Please don't be confused, they are different.
• "Tangent" is sometimes considered a brother or sister of sine and cosine. In some sense, this is true. When it was introduced by the Danish mathematician Thomas Fincke in 1583, he named it "tangens," which means "touching" in Latin.

So, we have a bay, a word which sounds similarly and a touch. These are the origins of the terms sine, cosine, and tangent. Today these words don't convey any lexical meaning for the majority of people, they are considered like words of the math language.

Definition of cos, sin, and tan

Depending on the reader's profile we could provide a different definition for the heroes of the current article. But the most classical explanation is usually provided through the definition of the properties of a right triangle. Let's introduce a naming schema:

Let's define a right triangle with a, b, and c sides, and A, B, and C as angles. Then the terms are defined as functions of the A or B angles. Please pay attention that they are never defined for the C angle, since C is the right angle.

1. Sine of A is defined as a/c, or the opposite a side divided by the hypotenuse c.
2. Cosine of A is b/c, or the adjacent b side over the hypotenuse c.
3. Tangent of A is a/b, or the opposite a over the adjacent b side.

In a similar way, the functions could be redefined for the B angle.

There are other definitions that rely on the circles, limits and sums, and others. Let's just not dive into those. We are not lazy to define them, we are trying not to over complicate the article. If the reader still wants to see other definitions then they contact us and ask to write an article. It will be immediately done. We promise ))

Other related functions

The reader for sure already knows that sine, cosine and tangent are not the only functions. There are others such as secant, cosecant, cotangent and the somehow forgotten exsecant. These functions indeed are used and cannot be ignored. Their explanation would require more time.

The writer of the article just prefers to say that they are defined through the already described 3 functions.

Interesting facts

Maybe this is how new words appear in every language. Mathematics, as the reader knows, also has a language of its own, and it's quite interesting that cosine, sine, and tangent appeared simply because they sounded relevant.

• We have already provided it but the word "sine" exists because of a translation error. Arabic equivalent "jaib" appeared as misread of a transliterated Indian word.
• "Cosine" word had no any etimology at the moment when it was invented, it is just sounds familiar to the "cosine".
• "Tangens" is a "touch" in Latin. Indeed, the tan function is often defined throught a square triangle and circle which it touches.

Impact and legacy

"Sine," "cosine," and "tangent" are now essentially universal across all languages. If you study mathematics, science of engineering or anything that requires a skill to calculate values then you pronounce these words similarly in any language. Remember, mathematics language is a separate formal language.

You can research space, process images, build quantum computers, assemble electrical circuits, predict economical cycles. These terms will be always with you.