Ok, here is where things get weird: rather than a flat plane, we are looking at drawing figures on the outside of a sphere, where the sum of the interior angles of a triangle can add up to more than 180 degrees. Basically, pure anarchy prevails here…or so many thought until mathematicians took a look and produced some laws of spherical trigonometry. But before we get ahead of ourselves, let’s take a look at spherical triangle ABC:
Rather than having straight edges, a spherical triangle is made up of three intersecting circles, whose centers are at the center of the sphere, (0, 0, 0) in this case (also known as a great circle). This fact means a shift in how we think about side lengths. Rather than arc AC being referred to by its length, because it is part of a great circle, it can be referred to by an angle and I will be doing this in radians for the remainder of the page. Although there is law of cosines and law of sines for spherical trigonometry it looks different because everything is done with angles. Based on a zoomed in picture of the sphere above, we have:
Now, the proof of this is extensive and painful, therefore I have attached it as a pdf here (to be added later) rather than including it on the page. If you have not already done so, read The Power of Tangency and Rules of Traditional Trigonometry before engaging with the proof.
Below, research some of the weird properties that emerge when using spherical trigonometry:
- Changing Relationships: Shockingly, a spherical triangles has angles that can add up to more than 180 degrees. Have students investigate some of the changing relationships.