Angles and Radian Measure

By the end of this lesson, you will be able to define angles in standard position, find coterminal angles, compute arc length and sector area in radians, and convert fluently between degrees and radians.

Angles in Standard Position

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis.

The terminal side determines the angle. Positive angles go counterclockwise; negative angles go clockwise.

Coterminal Angles

Angles are coterminal if they share the same terminal side — they differ by multiples of $360°$ (or $2\pi$ rad).

$$\theta + 360°n \quad (n = \pm1, \pm2, \ldots)$$

Example: $390°$, $30°$, and $-330°$ are all coterminal.

Finding coterminal in $[0°, 360°)$: add or subtract $360°$ until in range.

Reference Angles

The reference angle $\theta'$ is the acute angle between the terminal side and the $x$-axis ($0° < \theta' \leq 90°$).

Quadrant	Reference angle
I	$\theta' = \theta$
II	$\theta' = 180° - \theta$
III	$\theta' = \theta - 180°$
IV	$\theta' = 360° - \theta$

Arc Length Formula (Radian)

For a circle of radius $r$ and central angle $\theta$ (in radians):

$$s = r\theta$$

This simple formula is why radians are the preferred unit in calculus.

Sector area: $A = \frac{1}{2}r^2\theta$

Angular vs. linear velocity:

$$v = r\omega$$

where $\omega$ is angular velocity (radians/second) and $v$ is linear velocity.

Example 1 — Coterminal Angles

Find two positive and two negative angles coterminal with $\theta = 5\pi/6$.

Positive: $5\pi/6 + 2\pi = 17\pi/6$ and $5\pi/6 + 4\pi = 29\pi/6$

Negative: $5\pi/6 - 2\pi = -7\pi/6$ and $5\pi/6 - 4\pi = -19\pi/6$

Example 2 — Reference Angle