Spherical law of cosines

The spherical law of cosines is a way to measure the distance between two points on the surface of a sphere, like the Earth. Imagine you and your friend wanted to measure the distance between your houses, but instead of living on a flat street, you both live on opposite sides of a giant ball.

First, you both need to agree on a starting point, like the North Pole. Then, you start walking in opposite directions around the ball until you each reach your respective homes. The spherical law of cosines tells you the shortest distance, or great-circle distance, between your homes, even if you didn't walk directly toward each other.

The formula for the spherical law of cosines looks like this:

cos(c) = cos(a) * cos(b) + sin(a) * sin(b) * cos(C)

But don't worry, you don't need to know algebra to understand it.

Let's use some pretend numbers to explain. Say you and your friend live 100 miles away from each other on the surface of the Earth. The angle between the two points (C) is 90 degrees because you're on opposite sides of the ball.

Now let's say you start at the North Pole (point A) and your friend starts at the equator (point B). The angle between point A and the equator (a) is 90 degrees, and the angle between point B and the equator (b) is also 90 degrees.

Using the formula, we can solve for the great-circle distance between your homes (c):

cos(c) = cos(90) * cos(90) + sin(90) * sin(90) * cos(90)
cos(c) = 0 + 1 * 1 * 0
cos(c) = 0

Now we need to solve for c, so we can use the inverse cosine function (cos^-1) to find the answer:

cos^-1(0) = 90 degrees

This means that the great-circle distance between your homes is 90 degrees, or one-quarter of the circumference of the Earth.

In simpler terms, the spherical law of cosines helps us figure out how far away two points are from each other when they're on a giant ball, like the Earth. It's like using a math formula to find your way to your friend's house on the other side of the world!