Walk around a circle and discover that sin and cos are just coordinates
One circle unlocks all of trigonometry. Every sine value, every cosine value, every identity — they all live on the unit circle. The unit circle is a circle with radius 1 centered at the origin. Every point on it can be written as (\cos\theta, \sin\theta), where θ is the angle measured from the positive x-axis.
This means cosine is just the x-coordinate and sine is just the y-coordinate of a point on the circle. That one insight unlocks all of trigonometry: sign changes in different quadrants, the Pythagorean identity \sin^2\theta + \cos^2\theta = 1, and the values of sin and cos at every key angle.
In this lesson, you'll walk around the unit circle angle by angle — from 0° to 360° — placing points and reading off their coordinates. By the end, sin and cos won't be abstract formulas; they'll be positions on a circle you can see and touch.