Comments on: Trigonometry lesson

By: Jonathan

Jonathan — Mon, 23 Apr 2012 05:20:38 +0000

In my biased opinion, even more remarkable than the three or four words for hill is the case where the same word contributes twice, as in Pendle Hill.

By: Jon Awbrey

Jon Awbrey — Sun, 22 Apr 2012 16:04:18 +0000

Thanks very much, this is fascinating. I remember reading somewhere that the Greeks used what we now call “double sines”, but I had always thought the word had something to do with the Latin sine for “without”, connoting a measure of deviation, and thus related to sin. Co-incidentally, here is a place where I found myself pondering the intrinsic meaning of trigonometric terms in the process of asking myself what would be the logical analogue of a tangent functor. • Differential Analysis of Propositions and Transformations

By: Peter Cameron

Peter Cameron — Sun, 22 Apr 2012 14:33:51 +0000

Actually it would be the arithmetic mean of the distances from the external point to the two points of intersection with the circle.

By: Steven

Steven — Sun, 22 Apr 2012 14:20:53 +0000

The ‘secant’ would cut the circle in two points if extended in the other direction, but only one half of the line is used to calculate the length called secant. Maybe this could also be said to be the case with the tangent, since only the ‘half-tangent’ is used to calculate the length called tangent (from the point where it meets the circle to the point where it meets the extended radius)?

By: Peter Cameron

Peter Cameron — Sun, 22 Apr 2012 14:09:03 +0000

Which is another lesson in how meanings change. To me, a secant is a line meeting the circle in two points; if it has to be a length, it should probably be the distance between these two points, or just possibly the distance from the distinguished point on the secant to one or other point where it meets the circle…

By: Steven

Steven — Sun, 22 Apr 2012 13:59:39 +0000

You got the tangent of the angle by extending the tangent at the end of one radius to meet the other extended radius. The length along this extended radius from the centre of the circle to where the tangent intersects the radius is the secant. I think the name makes sense as the extended radius is a line which cuts the circle, so it is indeed part of a secant. There is a diagram here:

http://upload.wikimedia.org/wikipedia/commons/4/45/Unitcircledefs.svg