 So remember that Ptolemy's Alamages began with a table of chords in a circle. Now, given chords, subtending central angle theta and rho, Ptolemy also included theorems for computing the chords subtending the central angle theta minus rho, the chord of the difference, and the chords subtending the central angle one half theta, the chord of half the angle. However, we can go more simply using only the Pythagorean theorem Indian mathematicians could find the chord of one half theta, which relied on the half chord. The half chord became even more important with Islamic geometers who considered problems involving sundials. As the sun moves, the shadow of the nomen can be used to determine the time. Now the actual path of the sun across the sky is very nearly a circle. And so the line connecting the tip of the nomen to the sun's position traces out a cone. That comes from Apollonius' reinvention of the conic sections. And the plane of the ground cuts the cone. And so the tip of the shadow traces out a conic section. And one consequence of this is Islamic geometers became very interested in conic sections and developed a number of important results about them and using them. We'll take a look at some of those results elsewhere. Now, while we're used to thinking of sundials as objects on the ground with a vertical nomen, this isn't always practical. Only those nearby can actually see the sundial. And the sun will actually be blocked by buildings. So more practically, the nomen might be stuck on a wall and run horizontally. And so more people can see it. And if it's high enough, it won't be blocked by other buildings. Now, since the length of the nomen is fixed, this gives us two more lengths of interest. First, what we could call the length of the shadow of the horizontal nomen or the length of the reverse shadow of the vertical nomen. And similarly, if the nomen is perpendicular to a surface, it forms a right triangle. And this gives us two more quantities of interest. The hypotenuse of the shadow and the hypotenuse of the reverse shadow. In the 10th century, Nasser Adin described these lengths in relation to a circle. Suppose we have a horizontal nomen OA on vertical wall AE. Since the nomen has a fixed length, we can make OA the radius of the circle. Suppose the sun's angle of elevation is theta. The satellite cuts the circle at some point B, and GB will be the half-court of theta. AD will be the shadow, and OD will be the hypotenuse of the shadow. On the other hand, if we have a vertical nomen OC on ground plane CF, then HB is the half-court of the complementary angle, CF is the reverse shadow, and OF is the hypotenuse of the reverse shadow. And this gives us six values for any angle theta. The half-court, the half-court of the complement, the shadow, the reverse shadow, the hypotenuse of the shadow, and the hypotenuse of the reverse shadow. Since the length of the cord depends on the radius of the circle, we'll use capital SINE theta to refer to the length of the cord GB, where the radius is implied. Islamic Geometers recognize that, given the sine and shadow, the remaining lengths could be determined from the Pythagorean theorem, so they focus their efforts on finding these values. For example, suppose we want to find the sine and shadow of 30 degrees, and we'll assume a radius of 60, which Islamic Geometers usually did following the tradition of Ptolemy. And again, once we have the sine and shadow, we can also find the sine of the complementary angle, the reverse shadow, the hypotenuse of the shadow, and the hypotenuse of the reverse shadow. And let's round our answers to the nearest whole number. So let's go ahead and draw a picture. And remember, these come from the shadows of vertical or horizontal nomans. So we can make both of those the radii of a circle. So we have a central angle of 30 degrees, and the radius is 60. So this length OB is equal to 60. And that means this triangle OGB is a 30-60-90 right triangle, and so we know that GB is equal to 30 the sine. And now we have two sides of a right triangle, so we can use the Pythagorean theorem to find the third side, which is the sine of the complement, and we'll start calling it the complement sine. And the thing that's useful to notice here is that triangle OGB is similar to triangle OAD. And so we can use similar triangles. AD to OA is equal to GB to OG. We know that OA is the radius, so it's equal to 60. We found GB and OG. So we solve for the reverse shadow. We have another pair of similar triangles, OHB and OCF. And so we know that CF is to OC as BH is to OH. And since OC is the radius, and BH is equal to OG and OH is equal to GB, we can substitute in those values and solve for the length of the reverse shadow. And we can find the two hypotenuses by the Pythagorean theorem again. One last note. Indian mathematicians used the word shiva for the half-court. Arabic mathematicians treated this as a technical term and transliterated it, keeping the sound without worrying about the meaning. And in Arabic this becomes shiva. Now, the Islamic works on trigonometry began making their way into European consciousness during the Middle Ages, and so Robert of Chester, who lived around 1125, misread this as jaiib. So he could have done the same thing that the Arabic mathematicians did and treated this as a technical term and transliterated it as jaiib, which would have at least retained the etymological connection to the half-court. But rather than treating it as a technical term, he translated it, and this word jaiib in Arabic means bay, and in Latin this is sinus. And the complement of the sign is the cosinus.