Trigonometry lesson


This is not a lesson about trigonometry. But where did the names sine, cosine, and the rest of them come from?

First, there is no mystery about the prefix “co”. Each cofunction of an angle θ is the function of the complementary angle π/2−θ.

Also, tangent is easy to explain. Take a circle of unit radius, and take two radii making an angle θ with each other. For convenience I will suppose that θ lies between 0 and π/2. Now draw the tangent at the end of one radius and extend it until it meets the other one produced. How long is it? The answer of course is tan θ.

But a similar explanation for secant runs into trouble. Take the secant joining the ends of the two radii. How long is it? The answer is 2 sin(θ/2). Apart from the fact that we have twice a trigonometric function of half the angle, doesn’t this suggest that what we call sine should be secant? I don’t really understand why the term “secant” is applied to the reciprocal of cosine.

But the story of where “sine” comes from is a curious one. The information here is taken from The Golden Age of Indian Mathematics, by S. Parameswaran. The main goal of the book is to describe the remarkable achievements of Keralan mathematicians in the period 1350–1600, especially Madhavan (1340–1425) who, among other things, found and proved the power series for the sine, cosine and inverse tangent functions.

As an aside, why are these discoveries not better known? I think there are two reasons. First, the masters were rather secretive about their work, communicating it only to trusted students. Second, they were written on palm leaves, and the ravages of time and the warm damp Kerala climate mean that much has been lost.

Anyway, Parameswaran explains that the Sanskrit word jya, meaning “chord”, was also used for the length of the chord subtending a given angle in a circle of standardised radius (the 2 sin(θ/2) we met above), or, in the forms ardhajya or bhujajya, for half of this chord. An alternative form for jya was jiva, which was adopted by the Arabs and became jiba. This was later confused with the Arabic jaib, a bay or inlet, and when the Arabic texts were translated into Latin, the word jaib was translated as sinus with the same meaning. Parameswaran remarks,

Hence came the word sine, providing an extreme example of a mathematical term which is completely bereft of its etymological meaning.

(The Oxford Etymlogical Dictionary gives “bosom” as an alternative meaning of both jaib and sinus.)

This process of misunderstanding has happened often when two peoples with different language interact. Examples commomly occur with placenames. The names Bredon Hill and Torpenhow Hill, for example, have been formed by the juxtaposition of three or four words for “hill”. Closer to the case of “sine” is the name Spinis (“place at the thorn-bushes”) from Roman times, which became Speen (“place where wood-chips are found”) in the tongue of the Saxons. (And, to digress even further, this is Speen in Berkshire, not Speen in Buckinghamshire, where the type designer and letter cutter Eric Gill had his last home and workshop, in a compound which is now a music school – I passed it on a walk from Saunderton to Chesham last month.)

The Keralan interest in trigonometry is partly explained by the connection of spherical trigonometry with astronomy, and hence with astrology, the applied mathematics of its day; it was used not as a device for prophecy, but to choose appropriate dates for the many rituals of Brahmanical life, which begin before birth and continue for many years after death. As Parameswaran says,

Jyotis-sastra (science of celestial luminaries) … comprises two parts, a theoretical part and a practical part … The phases of the moon, solar and lunar eclipses, and variations in the movement of planets … belong to the theoretical part. Readings of horoscopes …, reckoning of muhurtams (auspicious moments) etc. fall within the scope of the [practical] part.

The picture above, by the way, has nothing to do with trigonometry; it is a bank in Lisbon.

About Peter Cameron

I count all the things that need to be counted.
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7 Responses to Trigonometry lesson

  1. Steven says:

    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:

    • 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…

      • Steven says:

        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)?

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

  2. Jon Awbrey says:

    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

  3. Jonathan says:

    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.

  4. Shaun says:

    Thanks for the intriguing post! While I am also one of those who rarely do, it is always interesting to stop and think about where names come from. They often have very interesting histories! I can honestly say that I have never given these common trigonometry terms a second thought. I guess I always get way too preoccupied with actually solving the problems! Thanks again.

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