Unit Circle
TrigonometryThe 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 (the point where the $x$ and $y$ axes cross). It is a powerful tool for understanding sine and cosine at any angle.
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 $(-1, 0)$. These coordinates tell us cos and sin directly.
Key Insight
Because the radius is $1$, every point on the circle is $(\cos(\text{angle}), \sin(\text{angle}))$. The unit circle lets you read off trig values just by looking at coordinates.
Definition
The unit circle is the circle $x^2 + y^2 = 1$. For an angle $\theta$ measured counterclockwise from the positive $x$-axis, the terminal point is $(\cos(\theta), \sin(\theta))$. This definition extends sine and cosine to all real angles, not just acute ones.
Example
At $\theta = 2\pi/3$ ($120^\circ$): $x = \cos(120^\circ) = -1/2$, $y = \sin(120^\circ) = \sqrt{3}/2$. The point $(-1/2, \sqrt{3}/2)$ lies on the unit circle because $(-1/2)^2 + (\sqrt{3}/2)^2 = 1/4 + 3/4 = 1$.
Key Insight
The unit circle unifies all the special angle values ($30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $\ldots$) into a single picture. Memorizing the unit circle means knowing exact trig values instantly for all multiples of $30^\circ$ and $45^\circ$.
Definition
The unit circle $S^1 = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ is a compact Lie group under complex multiplication: $e^{i\theta} \cdot e^{i\phi} = e^{i(\theta+\phi)}$. The characters of $S^1$ are $e^{in\theta}$ for integer $n$, which form a complete orthonormal basis of $L^2(S^1)$, the foundation of Fourier series.
Example
The arc length of a sector of the unit circle with central angle $\theta$ equals $\theta$ radians. This is the definition of radian measure and explains why $d/dx[\sin(x)] = \cos(x)$ only when $x$ is in radians.
Key Insight
$S^1$ is the simplest example of a compact Lie group. 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.