Radians

Start at the centre of a circle. A radius joins the centre to the edge. An arc is part of that curved edge.

Take an arc as long as the radius, then join its two ends to the centre. The angle between those radii is one radian.

Call the arc length $s$, the radius $r$ and the central angle $\theta$ (theta). Count how many radius-lengths fit along the arc:

$\theta=\frac{s}{r}\quad\text{radians}$

For example, an arc of 6 cm on a circle of radius 3 cm gives $\theta=6/3=2$ rad. The lengths must use the same unit; their units cancel. The result measures an angle, not a length.

Make that circle twice as large without changing the angle: the radius and arc both double, so $s/r$ stays the same. In questions, an arc subtends an angle when its endpoints make that angle at the stated point.

The circumference, the whole distance around the circle, is $2\pi r$. Dividing by $r$ gives $2\pi$ radians in a full turn, so $\pi\text{ rad}=180^\circ$.

One degree is therefore $\pi/180$ radians. One radian is $180/\pi$ degrees, about $57.3^\circ$.

Useful angle values are $90^\circ=\pi/2$ rad, $180^\circ=\pi$ rad and $360^\circ=2\pi$ rad. For 45° or 60°, divide the half-turn by 4 or 3.

In terms of π means keep $\pi$ in the answer. If a decimal is requested, evaluate only at the end. An angle written as a multiple of $\pi$ normally has rad omitted, not a degree sign.

The sine, cosine and tangent keys relate an angle to ratios of side lengths; their inverse keys recover an angle from a ratio. Use RAD mode when working with radians. Multiplication such as $30\pi/180$ does not depend on angle mode.

A quick mode check is $\sin(\pi/6)=0.5$. Use the calculator's $\pi$ key, not 3.14.