I am teaching the foundations of trig and find it a bit weird that tangent is called that.

I've never questioned it before, but what I keep finding online is the phrase 'tangent of an angle'.

Is anyone able to explain, maybe using some visual intuition, why we call it the tangent of an angle? Especially in the context of the unit circle.

Does it relate to the definition of a tangent to a curve?

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    $\begingroup$ See The Etymology of Trig Functions. $\endgroup$ – J.-E. Pin Nov 23 at 10:43
  • $\begingroup$ It seems like Doug M's reasoning is correct. $\endgroup$ – KM101 Nov 23 at 10:43
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    $\begingroup$ In some places tg is short for tangent. $\endgroup$ – Henricus V. Nov 23 at 19:17
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    $\begingroup$ There are two questions here, (i) why is it called tangent and (ii) why is the standard abbreviation $\tan$. Regarding the latter, presumably ta is a bit ambiguous thank you and tang would be sued by General Foods Corporation. $\endgroup$ – copper.hat Nov 23 at 20:35
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    $\begingroup$ See english.stackexchange.com/questions/40364/… $\endgroup$ – Carmeister Nov 24 at 20:23

I can't say for certain that this is the origin of the name of the function.

But if you construct a line that is tangent to the unit circle, using either of these constructions:
then the red segment has a measure of $\tan x$.

In the second figure, the green segment has measure $\cot x$.

  • $\begingroup$ And to answer the follow-up question, the length of the horizontal segment from the origin to the base of the red segment is the secant of x, just as that line is a secant to the circle. $\endgroup$ – Matthew Daly Nov 23 at 11:38
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    $\begingroup$ Just for clarification: what is "x" in your diagram? $\endgroup$ – rprospero Nov 23 at 20:56
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    $\begingroup$ In your second figure, the red segment extends all the way from the x-axis to the y-axis, but the part of it to the left of the tangent-point is partly hidden by the green segment. So this is a bit confusing. I'm assuming that the red segment is only intended to extend from the x-axis to the tangent point? $\endgroup$ – ruakh Nov 23 at 22:39
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    $\begingroup$ The red segment is supposed to run from the point of tangency to the x axis and the green segement from the point of tangency to the y axis. $\endgroup$ – Doug M Nov 24 at 0:56
  • $\begingroup$ @rprospero x is the angle, PI/4 $\endgroup$ – Perdi Estaquel Nov 25 at 0:04

This is not an explanation why the word tangent is used (I think the other answers nicely do this), but a historical digression.

The first occurence of a mathematical symbol as denotation of the tangent-function I have personally seen is in Leonhard Euler's "Introductio in Analysin Infinitorum, Volume 2" from 1748. This was written in Latin which at that time was the universal language of European scholars. The Latin word is "tangens" which is the present participle of the verb "tangere" and Euler abbreviated it by the symbol tang. In many modern languages the word "tangens" was replaced by "tangent" (English) or similar forms ("tangente" in French and Italian) , but for example in German and Russian the original form is used. Note that the "t" in the end of the word comes in via Latin declination.

Although the concept of tangent was known much longer (for certain in Arabic/Islamic mathematics, first formal use by Abū al-Wafāʾ who lived from 940 to 998), the word "tangens" was at first introduced by Thomas Fincke in his book "Geometria rotundi" from 1583.

From Euler's "Introductio in Analysin Infinitorum":

enter image description here

Edited: After a little "research" I found that the symbol tan for tangent was introduced long before Euler. In fact, it seems that Albert Girard introduced it in his work "Trigonométrie" from 1626. However, other sources say it was introduced by Edmund Gunter in 1624.

It may also be interesting that the Arabic word for tangent is "shadow" which is in use for more than 1000 years. enter image description here

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    $\begingroup$ In French, we use "tangente" as a noun ("la tangente d'une courbe", "la tangente d'un angle"). As an adjective, "tangent" and "tangente" are used, depending on the gender of the noun they characterize. $\endgroup$ – Taladris Nov 24 at 1:41
  • $\begingroup$ @Taladris Thank you for clarification, I edited the answer. It shows how poor my French is - which is very sad ... $\endgroup$ – Paul Frost Nov 24 at 1:52
  • $\begingroup$ What does the Latin word "tangens"/"tangere" translate to in English? $\endgroup$ – nick012000 Nov 24 at 14:50
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    $\begingroup$ @nick012000 tangere = to touch, tangens = touching $\endgroup$ – Hagen von Eitzen Nov 24 at 16:26

Although I hardly believe there is any particular reason for this, the following interpretation might be satisfactory.

How does one interpret/define $\sin x$ and $\cos x$ using a unit circle? Consider a unit circle (i.e. it has radius 1) with $O$ being it's centre. Suppose $A, B$ are two points on the circle such that $\angle AOB$ measures $x$ radian. Draw $BD\perp OA$. Then, $BD=\sin x$ and $OD =\cos x.$ Now you might be curious how $\tan x$ comes into this picture. Draw the tangent to this circle at $A,$ let it meet $OB$ at $C.$ Then, $CA=\tan x.$


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