Angle
An angle is the figure formed by two rays that share a common endpoint, measured in degrees or radians.
Formula
\text{Full rotation} = 360^\circ = 2\pi \text{ radians}
Definition
An angle is formed when two rays start at the same point, called the vertex, with the two rays as its sides; we measure angles in degrees or radians to describe how wide open they are, one full rotation equaling $360^\circ = 2\pi$ radians. The measure of an angle describes the amount of rotation from one side to the other, and in linear algebra terms, the angle between vectors $u$ and $v$ satisfies $\cos\theta = (u \cdot v) / (|u| |v|)$, with directed angles treated as equivalence classes of rotations modulo $2\pi$.
Example
When you open a book, the two covers form an angle; the corner of a square is a $90^\circ$ angle, and a full circle is $360^\circ$. Angle $ABC$ has vertex at $B$ with rays $BA$ and $BC$ as its sides; to convert between units, $90^\circ = \pi/2$ radians and $180^\circ = \pi$ radians. The angle between vectors $u = (1,0)$ and $v = (1,1)$ is $\arccos((1\cdot1 + 0\cdot1) / (1 \cdot \sqrt{2})) = \arccos(1/\sqrt{2}) = 45^\circ = \pi/4$ radians.
Key Insight
Degrees divide a full turn into $360$ equal parts, a choice made by ancient Babylonian astronomers, which is why there are $360^\circ$ in a circle. Radians are the more natural unit mathematically: the radian measure of an angle equals the arc length on a unit circle subtended by that angle, making it dimensionless and essential in calculus, since derivatives of trig functions rely on radians. The dot product formula for angles extends to any inner product space, defining "angle" in abstract settings like function spaces, which underpins Fourier analysis, where orthogonality of functions is analogous to perpendicularity of vectors.