Angles and radian measure

In school you measured angles in degrees, with a right angle being $90^\circ$ and a full revolution $360^\circ$ . The choice of $360$ is historical , Babylonian astronomers used a base-60 system. For mathematics, a different unit is more natural: the radian, defined intrinsically by the geometry of the circle.

Definitions

An angle is the figure formed by two rays sharing a common endpoint (the vertex). We extend this to a signed quantity: a positive angle is measured counterclockwise from an initial side; a negative angle is measured clockwise.

A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. So if a circle of radius $r$ has a central angle subtending an arc of length $s$ , then $\text{angle in radians} = \frac{s}{r}.$

A full revolution covers an arc of length $2\pi r$ , so a full revolution is $\frac{2\pi r}{r} = 2\pi\ \text{radians} = 360^\circ.$ Hence the master conversion: $\boxed{\pi\text{ rad} = 180^\circ \quad\Longrightarrow\quad 1^\circ = \frac{\pi}{180}\text{ rad}, \quad 1\text{ rad} = \frac{180^\circ}{\pi} \approx 57.296^\circ.}$

Arc length and sector area

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

Arc length: $s = r\theta$ .
Sector area: $A = \tfrac{1}{2} r^2 \theta$ .

Both formulas fail if $\theta$ is in degrees , you would need to first convert.

Why radians win

When you write $\sin x$ in radians, you get the clean formula $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ , false if $x$ is in degrees. All later calculus formulas , derivatives, Taylor series, integrals , are stated in radians. Memorise the conversion and switch to radians as your default unit now.

Standard angles in radians

Degrees	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$135^\circ$	$150^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
Radians	$0$	$\tfrac{\pi}{6}$	$\tfrac{\pi}{4}$	$\tfrac{\pi}{3}$	$\tfrac{\pi}{2}$	$\tfrac{2\pi}{3}$	$\tfrac{3\pi}{4}$	$\tfrac{5\pi}{6}$	$\pi$	$\tfrac{3\pi}{2}$	$2\pi$

You must know these conversions by heart.

Generalised angles

In Class XI we no longer restrict to $0 \le \theta \le 360^\circ$ . An angle can be:

Coterminal: $\theta$ and $\theta + 2\pi k$ (for $k \in \mathbb{Z}$ ) are coterminal , they end at the same position on a unit circle but represent different rotations.
Negative: angles measured clockwise.
Greater than $2\pi$ : more than one full revolution.

This freedom is what allows $\sin, \cos, \dots$ to be defined for every real number.

Worked examples

Example 1. Convert $75^\circ$ to radians.

$75^\circ = 75 \cdot \dfrac{\pi}{180} = \dfrac{5\pi}{12}$ rad.

Example 2. Convert $\dfrac{7\pi}{12}$ rad to degrees.

$\dfrac{7\pi}{12} \cdot \dfrac{180^\circ}{\pi} = 7 \cdot 15^\circ = 105^\circ$ .

Example 3. A circle has radius $14$ cm. Find the arc length subtended by an angle of $\dfrac{\pi}{6}$ rad.

$s = r\theta = 14 \cdot \dfrac{\pi}{6} = \dfrac{7\pi}{3}$ cm $\approx 7.33$ cm.

Example 4. Find the sector area of a circle of radius $10$ cm cut off by an angle of $60^\circ$ .

Convert: $60^\circ = \pi/3$ rad. Area $= \tfrac{1}{2} r^2 \theta = \tfrac{1}{2} \cdot 100 \cdot \dfrac{\pi}{3} = \dfrac{50\pi}{3}$ cm $^2 \approx 52.4$ cm $^2$ .

Example 5 (harder). If a wheel of radius $35$ cm makes $30$ revolutions per minute, find the linear speed of a point on the rim in m/s.

Angular speed: $30$ rev/min $= 30 \cdot 2\pi$ rad/min $= 60\pi$ rad/min $= \pi$ rad/sec.

Linear speed: $v = r\omega = 0.35 \cdot \pi \approx 1.10$ m/s.

Try it yourself

Convert to radians: $15^\circ, 240^\circ, -45^\circ, 720^\circ$ .
Convert to degrees: $\dfrac{\pi}{5}, \dfrac{11\pi}{6}, -\dfrac{2\pi}{3}, 3$ rad.
Find the arc length of a circle of radius $20$ cm subtended by an angle of $135^\circ$ .
A pendulum $1.5$ m long swings through an angle of $0.2$ rad. Find the arc traced by the bob.
Find the angle in radians and degrees subtended at the centre of a circle of radius $7$ cm by an arc of length $22$ cm.
Find sector area: radius $14$ cm, angle $\dfrac{2\pi}{5}$ rad.
Express $\dfrac{5\pi}{4}$ rad and $-\dfrac{7\pi}{6}$ rad in degrees.
A horse running at $20$ km/h on a circular track of radius $50$ m: find the angular speed in rad/s.
Convert $30^\circ 45'$ (degrees-minutes) to radians.
Two arcs of the same length subtend angles of $60^\circ$ and $75^\circ$ in two circles. Find the ratio of their radii.
A clock's minute hand has length $10$ cm. Find the distance moved by the tip from $10:00$ to $10:25$ .
The Earth's radius is about $6400$ km. Find the arc on the surface corresponding to a $1^\circ$ change in latitude.

Pitfalls / Tricks

Always check the units before using $s = r\theta$ , the formula needs $\theta$ in radians.
Negative and large angles are legal. $-\pi/3$ and $5\pi/3$ are coterminal: both end at the same point on the unit circle.
$1$ radian is not a "small" angle: it is about $57.3^\circ$ , slightly less than $60^\circ$ .
Insight. A radian is the dimensionless angle: it is a ratio of two lengths (arc to radius). This is why all calculus formulas involving angles want radians and refuse degrees.