The Unit Circle
The single most important picture in trigonometry — and the bridge between algebra and the study of periodic functions.
1. Definition and Equation
The unit circle is the circle of radius $1$ centred at the origin $(0,0)$ of the Cartesian coordinate plane.
Recall the general equation of a circle with centre $(h,k)$ and radius $r$:
$$(x-h)^2 + (y-k)^2 = r^2$$
Setting $(h,k)=(0,0)$ and $r=1$ gives the equation of the unit circle:
Every point $(x,y)$ on the circle satisfies this equation, and every solution of this equation lies on the circle.
Why radius 1?
The choice $r=1$ is deliberate. When we inscribe a right triangle inside the circle with the hypotenuse as a radius, the hypotenuse equals $1$. The basic right-triangle ratios then simplify dramatically:
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{1} = y, \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{1} = x$$
No division is needed — the coordinates are the trigonometric values. This is why the unit circle is the natural home for defining sine and cosine.