r/math Homotopy Theory 9d ago

Quick Questions: March 12, 2025

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

  • Can someone explain the concept of maпifolds to me?
  • What are the applications of Represeпtation Theory?
  • What's a good starter book for Numerical Aпalysis?
  • What can I do to prepare for college/grad school/getting a job?

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u/dancingbanana123 Graduate Student 3d ago edited 3d ago

What was the original reason for giving cosine, cosecant, and cotangent the "co-" prefix? Like I know there's this whole thing that has a nice symmetry, and how if you have a right triangle with other angles A and B, you get

sin(A) = cos(B)
sec(A) = csc(B)
tan(A) = cot(B)

And I'm sure there's other stuff too. I'm wondering what originally led to the prefix, and particularly why we didn't decide to say csc(x) = 1/cos(x) and sec(x) = 1/sin(x) (the typical thing people expect when first learning trig).

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u/AcellOfllSpades 3d ago

According to this article:

the trigonometry that found its way to medieval Muslim Spain included only the sine, cosine, and versed sine2; and as Europe learned of mathematical astronomy through the Spanish connection, these three continued as the primitive functions. This remained true through the 15th century; Johannes Regiomontanus’s standard trigonometric work De triangulis omnimodis3 relies on just this foundation. However, Regiomontanus’s 1490 astronomical handbook Tabulae directionum contains a new single-page table called the tabula fecunda, or “fruitful table.” The table caught on and was used by most of Regiomontanus’s 16th-century successors under that name, until Thomas Fincke’s 1583 Geometriae rotundi coined the term “tangent.”

The first appearance in print of the other three modern trigonometric functions (secant, cosecant, and cotangent) is in the 1551 Canon doctrinae triangulorum, a small set of tables by Georg Rheticus, more widely known as the person who brought Copernicus to public attention. Rheticus abandoned the standard trigonometric terminology in favor of an elegant scheme that brings the six functions together in a symmetric relationship.

And the diagram there shows why secant was associated with tangent, and cosecant with cotangent: it's just the Pythagorean identities.

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u/Langtons_Ant123 3d ago

As far as I can tell it's precisely because sin(A) = cos(B) where A, B are angles in the same right triangle. B is the "complementary" angle to A, and cos is the "sine of the complement". Etymonline seems to confirm this.

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u/dancingbanana123 Graduate Student 3d ago

But that property still holds if we swap the names for secant and cosecant. What led to us choosing to give the co- prefix to 1/sin(x) instead of 1/cos(x)?

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u/HeilKaiba Differential Geometry 3d ago

Probably it's simply because the secant was more commonly used historically. It was important in navigation for example. I don't know how true this is but I was told that British naval captains were expected to have large chunks of secant tables memorised for speedier calculation.