Why is the cosecant "co"?

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Why is the cosecant "co"?

Postby skeptical scientist » Mon Jan 07, 2008 6:29 am UTC

While preparing for tomorrows lecture on trig functions, I was moved to ponder the following question:

Why is the cosecant "co"? In light of the definitions csc=1/sin, and sec=1/cos, it seems that it would have been more natural to rename the secant and cosecant in the opposite fashion. I was wondering if anyone here could illuminate on why this was not so (hopefully before a student asks me tomorrow morning). :P

Thoughts? I found nothing helpful on Wikipedia or Mathworld.
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Re: Why is the cosecant "co"?

Postby LoopQuantumGravity » Mon Jan 07, 2008 6:47 am UTC

skeptical scientist wrote:While preparing for tomorrows lecture on trig functions, I was moved to ponder the following question:

Why is the cosecant "co"? In light of the definitions csc=1/sin, and sec=1/cos, it seems that it would have been more natural to rename the secant and cosecant in the opposite fashion. I was wondering if anyone here could illuminate on why this was not so (hopefully before a student asks me tomorrow morning). :P

Thoughts? I found nothing helpful on Wikipedia or Mathworld.


Because it transforms like basis vectors and is covariant? (I've been writing lecture notes on tensors SR and E&M for people who don't know anything about tensors, SR, or E&M! :D)

I would imagine that it comes from the geometric picture somehow, rather than the algebraic one. How, I don't know, but it's something to think about.
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Re: Why is the cosecant "co"?

Postby Cosmologicon » Mon Jan 07, 2008 6:48 am UTC

I think it's from the geometric definition, too, but I took the time to make a picture. Behold! The lengths in blue are sine, tangent, and secant, while the lengths in red are cosine, cotangent, and cosecant:
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Re: Why is the cosecant "co"?

Postby jestingrabbit » Mon Jan 07, 2008 6:58 am UTC

@cosmologican: Your picture sucks. Behold the greatest diagram humanity shall ever know!

*cue dramatic music*
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Re: Why is the cosecant "co"?

Postby skeptical scientist » Mon Jan 07, 2008 6:59 am UTC

I guess that could work, although I usually draw cos along the x-axis, and one could equally well draw sin along the y-axis, which puts them in the wrong part of the picture for their color.

But that's probably the reason, so thanks!

And I dislike both pictures for the same reason: theta looks like pi/4, so you get congruencies you wouldn't in general.
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Re: Why is the cosecant "co"?

Postby Cosmologicon » Mon Jan 07, 2008 7:11 am UTC

I think it's clearly less than pi/4 in my diagram. I originally had it smaller, but the cosecant wound up making the image really tall. So I made it as close to pi/4 as I could while still being obviously smaller: I also don't like it being some special angle.

@jestingrabbit: Hey, why so stingy with the labels? Why don't you give the points (0,1) and (1,0) a name too? That's clear things right up. ;-)

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Re: Why is the cosecant "co"?

Postby jestingrabbit » Mon Jan 07, 2008 7:14 am UTC

Cosmologicon wrote:@jestingrabbit: Hey, why so stingy with the labels? Why don't you give the points (0,1) and (1,0) a name too? That's clear things right up. ;-)


This is a diagram in the Euclidean plane, not the Cartesian plane, that's why smarty-pants!!

Edit: Then why the hell is there a 0 there... DAMN!!!!!
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Re: Why is the cosecant "co"?

Postby mike-l » Mon Jan 07, 2008 7:57 am UTC

Co functions are decreasing on (0,pi/2), non-co functions are increasing?
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Re: Why is the cosecant "co"?

Postby mikegoo » Tue Jan 08, 2008 4:04 am UTC

You could feed them some bull about making the pythagorean identities nicer...

sin2x +cos2x = 1
1 + cot2x = csc2x
tan2x +1 = sec2x
Notice how all the co's go together so nicely...

You could also spew forth about all of the basic derivatives of sec x, tan x, csc x, cot x being nicer since sec and tan have sec's and tan's...csc and cot have csc's and cot's.

Not to mention the fact that cos has an s and a c and so does sec...clearly it was meant to be!

Then you can laugh and claim you don't really know...the decision was made before they put you in charge and get a good laugh.

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Re: Why is the cosecant "co"?

Postby Token » Wed Jan 09, 2008 3:08 am UTC

Good god - how did I miss this thread? My one chance to show off some of my random knowledge about the history of maths, and I almost blew it. Still, better late than never.

*ahem*

The reason that sec is not the 1/sin is because, contrary to what is assumed in the OP, sec was never originally defined as 1/cos. How it was defined can be best illustrated by a diagram I just made in Paint.

sinsectan.JPG
sinsectan.JPG (10.2 KiB) Viewed 5137 times

AB is the chord, so naturally it is called the sine*. CD is the tangent line, so it is of course called the tangent. And OD is a secant line (the extension of a chord outside the circle), so it is called the secant. Simple enough when drawn like that. And, of course, the sine, tangent and secant of the complimentary angle are called the complimentary sine, complementary tangent and complimentary secant respectively (or cos, cosec and cot for short).

The diagram given above is a reshifting of this, but the original meaning of the lengths has been lost in the haste to get all the functions in one quadrant.


* OK, it's not so its not at all natural, but it's an interesting linguistic progression. The Hindu word for "chord" is "jya". This was then morphed into Arabic as "jiba", which isn't actually an Arabic word. Because of this, and the fact Arabic is usually written without vowels, there was no reason to later interpret "jb" as "jiba" and then get the Hindu "jya" when it was being translated into Latin in the 12th century. Instead, it was read as "jaib", which means "bay". This was then translated into Latin as "sinus", and so we get the English word "sine". Fascinating, huh?
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Re: Why is the cosecant "co"?

Postby Nimz » Wed Jan 09, 2008 9:16 am UTC

jestingrabbit wrote:
Cosmologicon wrote:@jestingrabbit: Hey, why so stingy with the labels? Why don't you give the points (0,1) and (1,0) a name too? That's clear things right up. ;-)


This is a diagram in the Euclidean plane, not the Cartesian plane, that's why smarty-pants!!

Edit: Then why the hell is there a 0 there... DAMN!!!!!

What's wrong with having an O there? You need a name for that point to name the line segments that have lengths sec θ and csc θ in your diagram.

Token's explanation is better, though, so you don't get the points you would have otherwise gotten from having the O.
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Re: Why is the cosecant "co"?

Postby jestingrabbit » Wed Jan 09, 2008 9:26 am UTC

Nimz wrote:What's wrong with having an O there? You need a name for that point to name the line segments that have lengths sec θ and csc θ in your diagram.


You don't need to label the lengths, but you do need to indicate that the 'axes' are crossing at right angles.

It probably seems like I'm disociating myself from the diagram, but I believe I lifted it from someone who lifted it from an MIT website.

Nimz wrote:Token's explanation is better, though, so you don't get the points you would have otherwise gotten from having the O.


Token's explanation is an actual explanation, rather than a diagram and a guess.

@Token: Incredible that the etymon of sine is jya! Good to know.

Edit: an even more content packed diagram, that's a little prettier too imo.

http://en.wikipedia.org/wiki/Image:Circle-trig6.svg
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Re: Why is the cosecant "co"?

Postby Sir_Elderberry » Sun Jan 13, 2008 5:40 pm UTC

I'm kind of out of my league on the math board, I know, but my understanding of this was aided by what my trig book labels as the cofunction identities.

In degrees, because division bars don't look good in type:

sin 90 - x = cos x
cos 90 - x = sin x
sec 90 - x = csc x
csc 90 - x = sec x
tan 90 - x = cot x
cot 90 - x = tan x

http://en.wikipedia.org/wiki/Cofunction
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Re: Why is the cosecant "co"?

Postby NathanielJ » Sun Jan 13, 2008 9:08 pm UTC

I'm kind of out of my league on the math board, I know, but my understanding of this was aided by what my trig book labels as the cofunction identities.

In degrees, because division bars don't look good in type:

sin 90 - x = cos x
cos 90 - x = sin x
sec 90 - x = csc x
csc 90 - x = sec x
tan 90 - x = cot x
cot 90 - x = tan x


That doesn't explain the "co" thing though, because those equations all work exactly the same even if you were to define sec x as 1/sin x and csc x as 1/cos x.

I think token wins the thread.
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Re: Why is the cosecant "co"?

Postby venkateshmrao » Thu May 09, 2019 8:09 am UTC

'jya' is a Sanskrit word. It is used for a chord of a circle/arc and also in Vedic mathematics for the 'sine'. It is also the Sanskrit word for bow string - since it does form a chord of an arc (i.e the bow)
The cosec is called 'vyutkrama-jya' which means inverted jya.

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Re: Why is the cosecant "co"?

Postby Eebster the Great » Thu May 09, 2019 2:14 pm UTC

Also, jaib means "breast," not "bay." The Latin translation sinus can mean either "breast" or "bay," but it was chosen as a translation of the former. Centuries later, mathematicians plotted the graph of y = sin x on Cartesian coordinates and noticed that it does kind of look like boobs.

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Re: Why is the cosecant "co"?

Postby DavCrav » Thu May 09, 2019 7:48 pm UTC

the 'co' stands for complementary. It's the sine/secant/tangent of the complementary angle (i.e., pi/2-angle).

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Re: Why is the cosecant "co"?

Postby Isaac Hill » Sun May 12, 2019 8:59 pm UTC

Sir_Elderberry wrote:I'm kind of out of my league on the math board, I know, but my understanding of this was aided by what my trig book labels as the cofunction identities.

In degrees, because division bars don't look good in type:

sin 90 - x = cos x
cos 90 - x = sin x
sec 90 - x = csc x
csc 90 - x = sec x
tan 90 - x = cot x
cot 90 - x = tan x

http://en.wikipedia.org/wiki/Cofunction
(emphasis added)

Having co- prepended to the 1/sin version means you get:
tan(x) = sin(x)*sec(x)
cot(x) = cos(x)*csc(x)
All the co-s are together without the messiness of division bars. I hope the OP's lecture was delayed 11 years so this can be included.
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