Is a sine something besides a function? A tangent is a line that touches a circle at 1 point and a secant is a line that touches at 2 points. Is there similar geometric object that’s defined as a sine? Bonus question: how do sine and secant functions relate to their geometric objects?
Thanks,
Rob
Not sure if I understand your question exactly, but hopefully this chart is of help: https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Unitcircledefs.svg/250px-Unitcircledefs.svg.png
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If I’m reading Wikipedia correctly, the sine function came up as a way to measure a chord of a circle. The word “sine” comes from the Latin word “sinus” meaning “bosom” because someone erroneously translated an Arabic term meaning “chord (of a circle)”.
A sine is a measure of how much of a unit line which makes a given angle with the x axis is projected onto the y axis. It is, therefore, the imaginary part of the Taylor series expansion of e[sup]θi[/sup] for some angle θ. That Taylor series expansion can, in fact, be an operational definition of sine, in that it’s relatively tractable numerically and has all the right behavior when you take derivatives and integrals of the infinite sum.
If you want to move even further away from the whole ‘function’ concept, sine is a perfectly good vector and can, in fact, be a normalized basis vector in an infinite-dimensional vector space comprised entirely of periodic functions, from which you can build any other function you want by taking sums of potentially infinite subsets, suitably rescaled. This is the heart of Fourier analysis and whole load of signal processing theory and practice.
If you plot the sine fumction of a series of numbers on the X axis vs the cosine of the same numbers on the Y axis, you get a circle.
Sine is the axial displacement (vs the centre) of a point on the circumference of a circle, for any given angle.
I don’t think that sine has the same type of multiple meanings that tangent and secant have.
Alternators (AC electrical generators) - which are typically a cylindrical (round) field armature spun inside a cylindrical stator- produce electricity as sine waves.
The image in standingwave’s post does sine nice. But tangent??
One of the prettiest things about the sine function to me is that if you plot the slope of a sine wave you get a cosine wave. If you plot the slope of a cosine wave you get a negative sine wave. Plot the slope of that …
Throw in some initial values plus stuff about having nice derivatives and you can show that these are the only two functions that have this relationship.
Sine and cosine. Made for each other. Up to change in sign. Get it, “sign”? Oh, well.
It’s not exactly a geometric object, but it is the length of a geometric object.
Draw a circle of radius 1 and two rays from the center of the circle. Two rays form an angle. The “sine” of that angle is the length of the shortest possible line segment that connects from one of the rays to the point where the other ray intersects the circle.
You could also say “the length of the line segment that is perpendicular to one of the rays and intersects the point where the other ray intersects the circle”.
For example, if the angle formed is 30 degrees, the line segment has length 0.5, hence the “sine” of 30 degrees is 0.5 .
P.S.
I must admit that Euclid, Pythagoras, et al, would not have looked at it this way. Here in the 21st century, we tend to think of fractions as numbers, but they would have thought of a fraction as a ratio. When we talk about the number ½, we think of a point on the number line which is halfway between zero and one, represented by the decimal 0.5 . But they would have thought of it as two line segments, one of which is exactly twice as long at the other. Hence, they would not have started by saying the circle has radius 1, they would have said a radius is a line segment connecting the center of the circle to the edge of the circle. So, from their perspective…
Construct a circle (of any size). The center of the circle is a point, which we call “c”. Construct a ray starting at c. The point where the ray intersects the circle is “a”. Line segment ac is known as a radius. Construct another ray from c. From point a, construct a line segment which is perpendicular to the second ray and intersects the second ray at a point we will call “b”. This gives us line segment ab. The two rays form an angle. The “sine” of that angle is the ratio of ab to ac.
Again, take for example the angle we call 30 degrees. They would have said that the measure of the angle is the ratio of that angle to a right angle, which is 1:3, and sine of that angle is the ratio of ab to ac, which is 1:2. From their perspective, the measure of an angle is a ratio and the sine of that angle is also a ratio. The connection between the ratios is determined by the geometric construction I described above.
P.P.S. This perspective about ratios explains why it was so hard for them to accept the existence of irrational numbers, which they would have called “incommensurable distances”. The believed that, given any two line segments, either one of the line segments could evenly divide the other or there had to be a third segment which would evenly divide both of the first two. Given ab which is larger than cd, either cd divides ab n times (in which case the ratio is n:1) or there must exist ef which divides ab n times and divides cd m times (in which case the ratio is n:m). It was embarrassing to them that when you bisect a right angle, you get an angle whose sine cannot be written as a ratio of whole numbers. In that example, when you follow my construction above, you get ab and ac but there is no way to write their ratio with whole numbers. We can narrow it down, e.g. the ratio of ab to ac must be less than 5:7 and greater than 9:13, but every ratio of whole numbers is either less than or greater than, never equal to, the sine of 45 degrees, sqrt(2)/2.
Earlier Greek geometers had difficulty with incommeasurate distances, but by the time of Euclid, they had come up with an alternate formulation, based on similar triangles, that worked. Of course, as with all things Euclid, it’s impossible to tell how much of that was Euclid’s own creation, vs. a compilation of discoveries made by others.
According to Larry Gonick, citing some Indian book on mathematics he doesn’t name, “sine” comes directly from an Indian word meaning “bowstring”.
Have a look here:
Interesting. Here’s the Wikipedia etymology:
I’m not sure I’d call that “directly”, but it is coming from Sanskrit.
Yes that chart really does show the tangent as the length of a tangent line and the secant as the length of a cutting line. I had always heard that sinus was the Latin word for bay, not that is was bosom, but maybe I heard the bowdlerized version. But that picture suggests that bosom might be a better term.
Oh, I see that now. I was looking at slope rather than length.:smack:
Well quoting wikipedia doesn’t really help here, it is wrong.
You left out the important part which is ALSO the part which would highlight the error.
Sine doesn’t mean bosom. That might be someone being childish about the graph of | sin (x) |… a joke edit of wikipedia ? who’d have thought ?
The Sanskrit word for half chord is jya-ardha. Now they wrote that in Arabic as sounds “jiba”, which coincidentally sounds like the arabic word for curve. The Arabs may have felt it meant “curves” not “half the chord”… Anyway so the translation to latin is to a shortened version of the latin sinus which means “a curve”. Well you should already recognise that as the name of body parts… due to their curved or sack like shape.
It doesn’t seem reasonable that mathematicians thought they were writing or reading “mammary gland” when they wrote jiba.
Nice, nice.
This is why teaching ins and outs of Renaissance music theory to most numerically confident adults, and temperaments in particular, is often difficult. Who says 3/2 is incommensurate with 2:1?
Let me just add that the original trig function was chord, the length of the chord between the two ends of the two radial lines defining the angle. It is actually twice the sine of half the angle. The reason it is less convenient than sine is the summation formula sin(x+y) = sin(x)cos(y)+cos(x)sin(y), which turns out to have deep theoretical consequences.
No, the wikipedia entry is essentially correct; at least, your explanation is more wrong.
It is almost certain that the Sanskrit word transliterated into Arabic “jiba” was not “jya” or “jya-ardha”, but a synonym for “jya”, namely “jiva” (both words mean “bowstring” in Sanskrit, referring to the chord stretching across an arc like a bowstring across a bow).
It’s not that the Arabic transliteration “jiba” sounds like the Arabic word “jayb” meaning “bay/bosom/pocket”, but rather that the two words look alike. So the more familiar existing word “jayb” got applied to the concept of the sine, displacing the original foreign (Sanskrit) loanword “jiva/jiba”.
Nobody said they did. I don’t know what folk etymology, if any, Arab mathematicians came up with to explain the apparent assignment of a word meaning “bay/bosom/pocket/fold” to the concept of “sine”, but AFAIK it had nothing to do with breasts.
