The Unit Circle and Radian Measure

In geometry, we use degrees, but in advanced mathematics, we use radians. A radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle. Because the circumference of a circle is $2\pi r$, a full rotation is $2\pi$ radians. This means $360^\circ = 2\pi$ radians, or more simply, $180^\circ = \pi$ radians. To convert from degrees to radians, multiply by $\frac{\pi}{180^\circ}$. To convert back, multiply by $\frac{180^\circ}{\pi}$. Using radians allows us to treat angles as real numbers, which is essential for higher-level functions.

The Unit Circle is a circle with a radius of $r = 1$ centered at the origin $(0,0)$. Its equation is $x^2 + y^2 = 1$. For any angle $\theta$ in standard position, the terminal side intersects the circle at a point $(x, y)$. In this specific circle, we define our primary trigonometric functions as: Cosine is the $x$-coordinate ($\\cos(\theta) = x$), Sine is the $y$-coordinate ($\sin(\theta) = y$), and Tangent is the ratio of the two ($\tan(\theta) = \frac{y}{x}$). This transition allows us to move from right-triangle trigonometry to circular functions that can handle any angle, including negative angles and those greater than $360^\circ$.

When using radians, calculating the distance along a curve (arc length) and the space inside a 'pizza slice' (sector area) becomes incredibly simple. The arc length $s$ is given by $s = r\theta$, where $\theta$ is in radians. The area $A$ of a sector is $A = \frac{1}{2}r^2\theta$. These formulas are more elegant than their degree-based counterparts because radians are naturally scaled to the radius. If you are given an angle in degrees, you must convert it to radians before using these specific formulas.