Angles in standard position and radian measure | Trigonometry

Angles as rotations

In trigonometry, an angle is a rotation. 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 is where the rotation ends.

Counterclockwise rotations give positive angles.
Clockwise rotations give negative angles.
You can rotate more than once around: $$400°$$ is one full turn plus $$40°$$ .

$$400°$$ and $$40°$$ are coterminal; so are $$-90°$$ and $$270°$$ .

Radian measure

Degrees are an ancient, arbitrary unit. Mathematicians prefer a unit that comes from the circle itself.

A full circle has circumference $2\pi r$ , which is $2\pi$ radius-lengths. So a full turn is $2\pi$ radians:

360° = 2\pi \text{ radians} \qquad\qquad 180° = \pi \text{ radians}

One radian is about $$57.3°$$ . When no unit is written, an angle is in radians.

Converting

Degrees	$$30°$$	$$45°$$	$$60°$$	$$90°$$	$$180°$$	$$270°$$	$$360°$$
Radians	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$

Arc length

Radians make arc length simple. Since one radian corresponds to an arc of length $$r$$ , an angle of $\theta$ radians corresponds to an arc $\theta$ times as long:

Practice

Quick check

How many degrees is $\pi$ radians?

Quick check

Which angle is coterminal with $$45°$$ ?