# The Seqed and Modern Trigonometry Exploration

Several of the problems in the Rhind papyrus compute the so called seqed of a pyramid. The Egyptians computed the ratio (half the base)/height.

In the image of the triangle here the seqed is actually the run over the rise of the smaller right triangle that makes up the left half of the image. Note that in this case the seqed measures the inverse of the slope (which would be the rise over run). Recall that for a right triangle we can define several trigonometric functions. the best known of those are:

- sin(x) = opposite / hypothenuse
- cos(x) = adjacent / hypothenuse
- tan(x) = opposite / adjacent
- cotangent(x) = adjacent / opposite

So for the small right triangle that takes up half of the larger triangle - which can be thought of as a cross section of a pyramid - we can look at the cotangent and notice that it is exactly the seqed computed by the ancient Egyptians.

cotangent(x) = adjacent / opposite = half the base / height = seqed

So what exactly is the seqed (or if you prefer the cotangent) measuring? It gives a ration of the base versus the height and hence gives a measuer of the steepness of the triangle. And because the triangle really represents the pyramid, it measures how steep the pyramid is.