Extra-Terrestrial Trigonometry

How did elementary school students discover the sine, cosine and tangent?
by Daniel Wolf-Root

We live on a sphere (more or less), my students told me. If that’s the case, then how do we find the distance between two points on a sphere? After many students discovered a method for doing this along the surface of a sphere, Ryan T. asked: what about the distance through the center? This reduces to finding the length of the chord formed between two points on a circle, given the radius of the circle and the measure of the corresponding central angle. Ryan’s Question is the question that Hipparchus and Ptolemy asked, and has been dubbed “The Fundamental Problem of Trigonometry.” How could we solve it?


Immediately, Molly made the following conjecture: we can use the Pythagorean Theorem. Then Alex came up with the following procedure: divide the circumference by pi to find the diameter; divide that in half to get the radius; apply Molly’s conjecture. But then Ryan H. produced infinitely many counterexamples: one for any value of the central angle other than a right angle; so the Pythagorean Theorem can’t be applied except in a simple case where we can already find the chord length. Delaine then suggested to draw the altitude of the triangle from the central angle, thus forming two right triangles (so we can apply the Pythagorean Theorem) and two segments of the chord, each with the same length. This will bisect the central angle, forming two angles, each with half the central angle’s measure. But then Max and Isabella wondered: what’s that new distance? If we know the radius and want to find the chord’s length, then we must have this other distance in order to apply the Pythagorean Theorem. Delaine conjectured that this length is half the chord’s length. However, Joseph said this works only for a central angle of 90 degrees, in which case we have two right isosceles triangles, which is the special case mentioned above. Finally, a number of people started asking what amounted to: If you’re given two lengths and one angle or two angles and one length of a triangle, is there a way to deduce the other angles and lengths?

Thus the discussion ended and students investigated the ideas they wanted to in their small groups. After the break, each group presented its work. We summarized each discovery of the presentations. Basically, the students asked or discovered a large fraction of high school geometry in these presentations: figured out the ideas behind the “triangle congruence shortcuts” (ASA, SSS,that SSA doesn’t work, etc.); deduced properties of isosceles, right, equilateral triangles, triangles inscribed in circles, some ideas about infinity, and that ratios of the Max-Isabella length and half the chord length are involved. So we defined Opposite side from central angle and Adjacent side from central angle. From here it was a small step for the students themselves to define the sine, cosine and tangent ratios.

Daniel Wolf-Root

Daniel Wolf-Root is an Elementary Division instructor and teaches mathematics at El Cerrito High School. He has taught ATDP math classes for the past three years.