Angles, Radians, and Coterminal Angles, precalculus lesson

A full turn split into $360$ equal parts gives the degree, a convenient but arbitrary choice. The radian comes from the circle itself. It measures an angle by how many radius lengths of arc it cuts off. Both describe the same rotations, and calculus needs radians, so it helps to switch between them quickly.

Key terms

Angle

Definition An angle is formed by two rays with a common endpoint, the vertex. In trigonometry an angle is a rotation of a ray about the vertex, from a starting position called the initial side to a final position called the terminal side. A counterclockwise rotation has positive measure and a clockwise rotation has negative measure. A rotation may go around more than once, so the measure of an angle is not limited to one turn.

In plain words Picture a clock hand that starts on the initial side and turns. How far it turns and in which direction is the angle.

Example The minute hand of a clock turns clockwise. In $15$ minutes it sweeps a quarter turn, and in $90$ minutes it sweeps one and a half turns. Both angles have negative measure because the rotation is clockwise.

Degree

Definition One degree, written $1^\circ$ , is the measure of a rotation through $\frac{1}{360}$ of a full counterclockwise turn. A full turn is $360^\circ$ , a half turn or straight angle is $180^\circ$ and a quarter turn or right angle is $90^\circ$ . An acute angle measures between $0^\circ$ and $90^\circ$ , and an obtuse angle between $90^\circ$ and $180^\circ$ . Parts of a degree are written as decimals or in minutes and seconds, where $1^\circ = 60'$ and $1' = 60''$ .

In plain words A degree is one of $360$ equal slices of a full turn.

Example The minute hand sweeps $-90^\circ$ in $15$ minutes and $-540^\circ$ in $90$ minutes. A surveyor’s reading of $47^\circ 15' 36''$ converts to decimal degrees as

47 + \frac{15}{60} + \frac{36}{3600} = 47.26^\circ.

Standard position

Definition An angle is in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive $x$ -axis. The angle lies in the quadrant that contains its terminal side. If the terminal side lies on a coordinate axis, the angle is called quadrantal and lies in no quadrant.

In plain words Standard position fixes where every angle starts, so the angle is described completely by where its terminal side ends up.

Example A few angles in standard position.

Angle	Rotation	Terminal side
$120^\circ$	one third of a turn counterclockwise	Quadrant II
$-45^\circ$	one eighth of a turn clockwise	Quadrant IV
$270^\circ$	three quarters of a turn counterclockwise	negative $y$ -axis, quadrantal
$400^\circ$	one full turn and $40^\circ$ more	Quadrant I

Radian

Definition Place the vertex of an angle at the center of a circle of radius $r > 0$ . As the ray turns from the initial side to the terminal side, the point where it crosses the circle travels a distance $s$ along the circle, counting every full turn. This $s$ is the arc length the angle cuts off, and the radian measure of the angle is

\theta = \frac{s}{r},

taken negative for a clockwise rotation. The ratio is the same for every circle centered at the vertex, because the arc cut off by a fixed angle grows in proportion to the radius. An angle of $1$ radian cuts off an arc exactly as long as the radius. A full turn cuts off the whole circumference $2\pi r$ , so it measures $2\pi$ radians. Radian measure is a ratio of two lengths and has no unit, so an angle written without a degree sign is understood to be in radians.

In plain words A radian counts radius lengths along the arc. Lay the radius around the circle like a flexible ruler and it fits $2\pi$ times, a little more than $6$ .

Example On a circle of radius $5$ centimeters an angle cuts off an arc of $12.5$ centimeters. Its measure is

\theta = \frac{12.5}{5} = 2.5 \text{ radians.}

On a circle of radius $10$ centimeters the same angle cuts off $25$ centimeters, and the ratio is still $2.5$ .

Converting units

Definition A half turn measures both $180^\circ$ and $\pi$ radians, so $\pi \text{ radians} = 180^\circ$ . To convert,

\begin{aligned} \text{radians} &= \text{degrees} \cdot \frac{\pi}{180} \\[0.8em] \text{and} \quad \text{degrees} &= \text{radians} \cdot \frac{180}{\pi}. \end{aligned}

In particular $1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57.30^\circ$ and $1^\circ = \frac{\pi}{180} \approx 0.01745$ radians.

In plain words Multiply by $\frac{\pi}{180}$ to go to radians and by $\frac{180}{\pi}$ to go back. Use degrees for everyday angles and radians in calculus, where the special limit of $\frac{\sin x}{x}$ as $x \to 0$ equals $1$ only when $x$ is in radians.

Example The common angles are worth knowing by heart.

Degrees	Radians	Part of a full turn
$30^\circ$	$\frac{\pi}{6}$	$\frac{1}{12}$
$45^\circ$	$\frac{\pi}{4}$	$\frac{1}{8}$
$60^\circ$	$\frac{\pi}{3}$	$\frac{1}{6}$
$90^\circ$	$\frac{\pi}{2}$	$\frac{1}{4}$
$120^\circ$	$\frac{2\pi}{3}$	$\frac{1}{3}$
$135^\circ$	$\frac{3\pi}{4}$	$\frac{3}{8}$
$150^\circ$	$\frac{5\pi}{6}$	$\frac{5}{12}$
$180^\circ$	$\pi$	$\frac{1}{2}$
$270^\circ$	$\frac{3\pi}{2}$	$\frac{3}{4}$
$360^\circ$	$2\pi$	$1$

Going the other way,

\frac{7\pi}{4} \cdot \frac{180}{\pi} = 315^\circ.

An angle of $4$ radians is about $229.18^\circ$ , and since $\pi \approx 3.14 < 4 < \frac{3\pi}{2} \approx 4.71$ it lies in Quadrant III.

Coterminal angle

Definition Two angles in standard position are coterminal when they have the same terminal side. Angles $\alpha$ and $\beta$ are coterminal exactly when

\begin{aligned} \beta &= \alpha + 360^\circ k \\[0.4em] \text{or, in radians,} \quad \beta &= \alpha + 2\pi k \end{aligned}

for some integer $k$ . Every angle is coterminal with exactly one angle $\theta$ with $0^\circ \le \theta < 360^\circ$ , or $0 \le \theta < 2\pi$ , found by adding or subtracting full turns until the measure lands in that range.

In plain words Extra full turns, in either direction, bring the terminal side back to the same place. Coterminal angles have different measures but end in the same spot.

Example Each angle below is reduced by whole turns.

Angle	Full turns added	Coterminal angle in one turn
$1000^\circ$	$-2$	$1000^\circ - 720^\circ = 280^\circ$
$-410^\circ$	$2$	$-410^\circ + 720^\circ = 310^\circ$
$\frac{17\pi}{4}$	$-2$	$\frac{17\pi}{4} - 4\pi = \frac{\pi}{4}$
$-\frac{5\pi}{3}$	$1$	$-\frac{5\pi}{3} + 2\pi = \frac{\pi}{3}$

Check yourself

Convert $225^\circ$ to radians and $\frac{5\pi}{12}$ radians to degrees.

Show the answer

In radians the first angle is

225 \cdot \frac{\pi}{180} = \frac{5\pi}{4}.

In degrees the second is

\frac{5\pi}{12} \cdot \frac{180}{\pi} = 75^\circ.

Find the angle between $0^\circ$ and $360^\circ$ that is coterminal with $765^\circ$ , and the angle between $0$ and $2\pi$ that is coterminal with $-\frac{13\pi}{6}$ .

Show the answer

Subtracting two full turns gives $765^\circ - 720^\circ = 45^\circ$ . Adding two full turns gives

-\frac{13\pi}{6} + 4\pi = \frac{11\pi}{6}.

In which quadrant does an angle of $2$ radians in standard position lie? About how many degrees is it?

Show the answer

Since $\frac{\pi}{2} \approx 1.57 < 2 < \pi \approx 3.14$ , the angle lies in Quadrant II. In degrees it is

2 \cdot \frac{180}{\pi} \approx 114.59^\circ.

Angles, Radians, and Coterminal Angles

Key terms

Angle

Degree

Standard position

Radian

Converting units

Coterminal angle

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus

Resources for this lesson

Unit circle explorer