A table of sines
Kamal Lodaya, The Institute of Mathematical Sciences, Chennai

A triangle has three sides. In a right-angled triangle the longest side is opposite the right angle, it is called the {hypotenuse}. If the hypotenuse has length 1, the length of the side opposite to one of the other angles is called the {sine} of that angle. The length of the remaining side adjacent to the angle is called its {cosine}. Figure 1 shows the sine of a 30 degree angle and a 45 degree angle. It is customary to write "sin A" for the sine of angle A and "cos A" for its cosine.
Why take the hypotenuse to be of length 1? Because then we can write Pythagoras's theorem for right-angled triangles as a beautiful equation:
(sin A)^2 + (cos A)^2 = 1.
In this article all angles will be measured in degrees, so we write "sin 30" for an angle measuring 30 degrees. Consider the sides opposite these angles. By the principle of similar triangles, these sides, when divided by the length of the hypotenuse, will have the same value.
For simplicity we will only consider angles from 0 to 90, although the ideas can be extended all the way to 360 and even to negative angles. (Try it if you find it interesting.)

The first step: similar triangles
In a right angled triangle, if one angle is 30^o, the other is 60^o, since the sum of the angles of a triangle is 180 degrees. Can you work out from the pictures in Figure 1 that sin 30 = cos 60, cos 30 = sin 60, and more generally that sin A = cos (90-A) and cos A = sin(90-A)? This means that if we make a table of sines, we get a table of cosines for free.
Now look at the picture for sin 30 more carefully. If the triangle is reflected about the horizontal axis as shown, the angle in the larger triangle doubles, so the larger triangle ABD is equilateral. This means that AB=BD and BC=CD=half of BD. 
Can you see that sin 30 = 1/2? If you use Pythagoras's theorem, can you now calculate that sin 60 = cos 30 is half the square root of 3, about 0.866? (Hint: two of the sides of triangle ABC are known to be AB=1 and BC=1/2 so you can calculate AC; also cos 30=AC/AB).)
Now look at the picture for sin 45. In this case both sides of the triangle are equal, so sin 45 = cos 45. From Pythagoras's theorem, both must be the square root of 1/2, about 0.707. (Hint: Use the same arguments as above).
The 2nd century Egyptian-Roman scientist Claudius Ptolemy of Alexandria (then in Greece), tried to calculate a table of sines in his book, which we know by its translation into Arabic as {Kitab al-majisti}, the "majestic book". You may ask why he did that, we will discuss that below.
By considering narrower and narrower triangles, can you conclude that sin 90 = cos 0 = 1? Also that sin 0 = cos 90 = 0, but Ptolemy had no zero. That is a later invention!
You can also observe that as an angle goes from 0 to 90, its sine keeps increasing, while its cosine keeps decreasing.
So now we know the values of the sines of some angles that we learn in school. But how do we make a table of all sines?

The second step: regular pentagons
Now we will begin to see the kind of cleverness that led the Arabic mathematicians to call Ptolemy's book "majestic". He derives the value of sin 36 by inscribing a five-sided regular pentagon (all sides are equal, we will assume length 1) inside a circle. This idea comes from the book "Elements" of the earlier Greek mathematician of Alexandria, Euclid, who lived around 300 BC.
We draw inside the pentagon a star, as shown in Figure 2. As a pentagon can be made up of three triangles, its angles must add up to 3 times 180, which is 540. As the five angles of the pentagon are equal, each must be 108. So in triangle ABC, angle ABC is 108. The two sides AB and BC are the same, so the other two angles BAC and BCA of the triangle are equal. So each is 36 degrees. Our aim is to calculate sin 36 which is the length of BF in the picture when the length of the hypotenuse AB is taken to be 1. We will first try to calculate the length of BD, marked {y} in the picture.
By symmetry of the star, we can also calculate that like angle BAD, angle ABD is 36. So the remaining angle ADB in the smaller triangle is 108. The triangles ABD and ABC are similar! We will soon use this.
Switching to triangle BCD, Since AC is a straight line, angle ADB + angle BDC must sum to 180. Since ADB=108, the angle BDC is 180-108=72. Also angle DBC is 108-36=72. This tells us that the sides BC and DC of triangle BCD are the same.
The length AC is AD+DC=AD+BC. We named the length of BD as y, so is the length of AD. We decided the length AB is 1, so is BC. Because triangles ABD and ABC are similar, the ratio BD/AB is the same as the ratio AB/AC. We substitute the known values in BD/AB=AB/AC. We get the equation y/1 = 1/(y+1), which reduced by cross-multiplying to y^2 + y = 1.
This is a quadratic equation which we learnt how to solve in school. It turns out that y is obtained as y=(sqrt(5)-1)/2. Taking the square root of 5 to be about 2.236, subtracting one and halving, we get a number known as the "golden ratio", around y=0.618.
Let us return to sin 36. Since F is the mid-point of AC, DF=AF-AD=AC/2-AD=(1+y)/2-y. Substituting, DF is around 0.191, which is cos 36. Using Pythagoras's theorem, we also obtain sin 36 is 0.588. So our table now has sin 36, and so also sin 54.

The third step: dividing into two triangles
Ptolemy now shows us another majestic technique: how to calculate the sine of (A+B) when we are given two angles A and B. In Figure 3 these angles are GOC and COD in the right-angled triangle OGD with hypotenuse OD set to 1. So GD=sin(A+B) and we want to find the length GD, which is broken into GF and FD by the clever construction shown in the picture using the right-angled triangles OCD and OCE.
From the triangle OCD, the length OC is cos B and CD is sin B. Now in triangle OCE with hypotenuse of length cos B, sin A=EC/OC=EC/cos B. So EC=sin A x cos B, and so also GF=sin A x cos B.
Again Ptolemy looks for similar triangles. OCE is similar to DCF. Why? Because angle FCO is A, so angle FCD is 90-A, just like angle OCE. So angle FDC is A, just like COE.
Now in triangle DCF with hypotenuse of length sin B, sin FCD=FD/CD=FD/sin B. But sin FCD is also given by sin FCD=sin (90-A)=cos A, so FD=cos A x sin B.
So what is GD=GF+FD? Substituting each term by what we have just computed, we get sin(A+B) = sin A x cos B + cos A x sin B. Perhaps you already knew this as an identity in trigonometry, but now you see why it is true!

More steps and a stumbling block
I hope you like the derivation of this beautiful equation. If this kind of thing interests you, you can try deriving from the next picture another one:
sin(A-B) = sin A cos B - cos A sin B.
In the picture we consider triangle OAB and set the hypotenuse OB to 1, so we want to calculate the length of side AB.
If you know something about sines and cosines of negative angles, another idea is to try to derive this subtraction identity by replacing B by -B in the addition identity.
So now from sin 36 we can get sin 72, from sin 30 and sin 45 we can get sin 75, from these two we can get sin 3, and then the sine of all multiples of 3. Here Ptolemy reaches a stumbling block. He cannot calculate sines of any angle which is not a multiple of 3! In particular he cannot calculate sin 1. The Greeks called this the problem of "trisecting an angle". Ptolemy worked out an approximation to sin 1 using other methods which we will not go into. The exact solution for the Greek problem was not found for centuries.

But why make a table of sines?
We saw how the scientist Claudius Ptolemy made a table of sines. Ptolemy was an astronomer. In the third century BC, the earlier Egyptian astronomer Eratosthenes of Alexandria had used sines to calculate the size of the Earth. Ptolemy used sines to calculate the distance to the Moon and the distance to the Sun. The last distance was not very accurate because the angles involved were very small and difficult to measure, but it is still a very impressive achievement for someone living two thousand years ago!
Going into details would take some effort. The basic idea is this: on the Earth we have a system of latitude and longitude. So we can sometimes think of the distance between two places as their difference in latitude (or longitude). Since Chennai (13 degrees North) and Kedarnath (31 degrees North) have almost the same longitude, we can say that the distance between them is 18 degrees of latitude. Similarly the position of the Sun, Moon and stars in the sky (the "celestial" sphere), can be described using "celestial" systems of latitude and longitude.
Spherical trigonometry is a part of mathematics which deals with spheres like the Earth, Moon and Sun, and which deals with distances by converting them to angles (when this is justified). Having a table of sines is extremely useful in doing these conversions. So in Ptolemy's time a lot of astronomical calculation was done using spherical trigonometry. A much more precise sine table was made by Indian astronomer Aryabhata from Patna in 499 CE. In fact, the Latin word "sine" (meaning a bay) comes from the translation of the Arabic word "jaib" (pocket or fold in cloth), which was the translation of the word "jya" (meaning the chord of a bow) used by Aryabhata.
Surprisingly, this has remained a common feature of science. Quite often problems in science can be converted to problems in mathematics.
Based on {Heavenly mathematics} by Glen Van Brummelen