Angle Measure

What is an Angle?

An angle measures rotation.

Start with a ray called the initial side. Rotate it around a fixed point. Where it lands is the terminal side.

The amount of rotation is the angle.

Direction Matters

• Positive angles: counterclockwise rotation
• Negative angles: clockwise rotation

The same position can be reached by rotating either direction.

Degrees

One way to measure angles.

A full rotation = 360°

Why 360? The Babylonians chose it because 360 has many divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12…

Radians

A more natural way to measure angles.

One radian is the angle where the arc length equals the radius.

Since circumference = $2\pi r$, a full circle has arc length $2\pi r$.

Dividing by the radius $r$: a full rotation = $2\pi$ radians.

Converting Between Degrees and Radians

The key relationship:

$\pi \text{ radians} = 180°$

Degrees to radians: multiply by $\frac{\pi}{180}$

Radians to degrees: multiply by $\frac{180}{\pi}$

Examples:

Standard Position

An angle is in standard position when:

• The vertex is at the origin
• The initial side lies on the positive x-axis

This gives us a consistent way to talk about angles.

Coterminal Angles

Coterminal angles share the same terminal side.

They differ by full rotations: add or subtract $360°$ (or $2\pi$).

Examples of coterminal angles:

• $30°$, $390°$, $-330°$ are all coterminal
• $\frac{\pi}{4}$, $\frac{9\pi}{4}$, $-\frac{7\pi}{4}$ are all coterminal

To find a coterminal angle between $0°$ and $360°$, keep adding or subtracting $360°$ until you’re in range.

Why Radians?

Radians make calculus formulas simple.

The derivative of $\sin(x)$ is $\cos(x)$ only if $x$ is in radians.

In degrees, you’d get an ugly constant: $\frac{d}{dx}\sin(x°) = \frac{\pi}{180}\cos(x°)$

When no unit is specified, assume radians.