Math, Better Explained: Learn to Unlock Your Math Intuition

by Kalid Azad

Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.

x is how far you traveled along a circle sin(x) is how high on the circle you are So sin(x)/x is the ratio of how high you are to how far you’ve gone: the amount of energy that went in an “upward” direction. If you move vertically, that ratio is 100%. If you move horizontally, that ratio is 0%. When something moves a tiny amount, such as 0 to 1 degree from our perspective, it’s basically going straight up. If you go an even smaller amount, from 0 to .00001 degrees, it’s really going straight up. The distance traveled (x) is very close to the height (sin(x)). As x shrinks, the ratio gets closer to 100% — more motion is straight up. Radians help us see, intuitively, why sin(x)/x approaches 1 as x gets tiny. We’re just nudging along a tiny amount in a vertical direction. By the way, this also explains why sin(x) ~ x for small numbers.