Unit Circle
The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles.
Formula
x^2 + y^2 = 1; \quad (x, y) = (\cos(\theta), \sin(\theta))
Definition
The unit circle is a circle with a radius of exactly $1$, centered at the origin, and it is a powerful tool for understanding sine and cosine at any angle. Formally, it is the circle $x^2 + y^2 = 1$; for an angle $\theta$ measured counterclockwise from the positive $x$-axis, the terminal point is $(\cos(\theta), \sin(\theta))$, extending sine and cosine to all real angles, not just acute ones. As the set $S^1 = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$, it forms a compact Lie group under complex multiplication, $e^{i\theta} \cdot e^{i\phi} = e^{i(\theta+\phi)}$, whose characters $e^{in\theta}$ form a complete orthonormal basis of $L^2(S^1)$, the foundation of Fourier series.
Example
At $0^\circ$, the point on the unit circle is $(1, 0)$; at $90^\circ$, it is $(0, 1)$; at $180^\circ$, it is $(-1, 0)$. At $\theta = 2\pi/3$ ($120^\circ$): $x = \cos(120^\circ) = -1/2$, $y = \sin(120^\circ) = \sqrt{3}/2$, and indeed $(-1/2)^2 + (\sqrt{3}/2)^2 = 1$. The arc length of a sector with central angle $\theta$ equals $\theta$ radians, the definition of radian measure that explains why $d/dx[\sin(x)] = \cos(x)$ only when $x$ is in radians.
Key Insight
Because the radius is $1$, every point on the circle is $(\cos(\text{angle}), \sin(\text{angle}))$, letting you read off trig values just by looking at coordinates, and the unit circle unifies all the special angle values ($30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $\ldots$) into a single picture. $S^1$ is the simplest example of a compact Lie group, and its representation theory via Fourier series is the prototype for harmonic analysis on more complex groups like $SU(2)$ and $SO(3)$, used in quantum mechanics and crystallography.