Math Core

In Geometry, trigonometry lived inside right triangles, so every angle was between $0^\circ$ and $90^\circ$ . To describe things that spin and repeat, like wheels, tides and sound waves, you need angles of any size, including negative ones. You also need a unit for angles that fits naturally with circles. That unit is the radian.

Angles in standard position

Picture a ray that starts along the positive $x$ -axis and rotates about the origin. The starting ray is the initial side, and the ray where it stops is the terminal side. An angle drawn this way is in standard position.

Rotating counterclockwise gives a positive angle.
Rotating clockwise gives a negative angle.
The rotation can go past a full turn, so angles like $450^\circ$ make sense: one full turn plus $90^\circ$ more.

An angle θ in standard position. The initial side lies on the positive x-axis, and the angle opens counterclockwise to the terminal side.

The axes split the plane into four quadrants, numbered I to IV counterclockwise from the upper right. An angle "lies in" the quadrant that contains its terminal side. An angle whose terminal side lands on an axis, like $180^\circ$ , is a quadrantal angle and lies in no quadrant.

Radian measure

Degrees split a full turn into $360$ equal parts, a number chosen long ago for convenience. Radians measure an angle using the circle itself.

Definition

Radian

A central angle of a circle measures one radian when it cuts off an arc whose length equals the radius of the circle.

The arc has the same length as the radius, so the central angle is 1 radian (about 57.3°).

How many radians make a full turn? The whole circumference is $2\pi r$ , which is $2\pi$ radius-lengths. So a full turn is $2\pi$ radians, and

360^\circ = 2\pi \text{ radians} \qquad\text{or}\qquad 180^\circ = \pi \text{ radians}.

That single fact gives every conversion. Radian measures are usually written with no unit at all: an angle of $\pi/3$ means $\pi/3$ radians.

Converting between degrees and radians

Since $\pi$ radians $= 180^\circ$ :

\text{radians} = \text{degrees} \cdot \frac{\pi}{180^\circ} \qquad\qquad \text{degrees} = \text{radians} \cdot \frac{180^\circ}{\pi}

Worked example: Converting in both directions

Convert $135^\circ$ to radians.
Convert $\dfrac{7\pi}{6}$ to degrees.
Convert $3$ radians to degrees, to the nearest tenth.

Solutions.

$135^\circ \cdot \dfrac{\pi}{180^\circ} = \dfrac{135\pi}{180} = \dfrac{3\pi}{4}$ .
$\dfrac{7\pi}{6} \cdot \dfrac{180^\circ}{\pi} = \dfrac{7 \cdot 180^\circ}{6} = 210^\circ$ .
$3 \cdot \dfrac{180^\circ}{\pi} = \dfrac{540^\circ}{\pi} \approx 171.9^\circ$ .

It pays to memorize the common angles, since they come up constantly for the rest of the unit.

degrees	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
radians	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$	$\pi$	$\dfrac{3\pi}{2}$	$2\pi$

Tip

Think of angles as fractions of a half turn. Since $\pi$ is a half turn, $\dfrac{\pi}{6}$ is one sixth of $180^\circ$ , which is $30^\circ$ . Likewise $\dfrac{5\pi}{6}$ is five of those, $150^\circ$ .

Coterminal angles

Angles that share a terminal side are coterminal. Adding or subtracting a full turn doesn't change where the terminal side lands, so

\theta + 360^\circ k \quad\text{or}\quad \theta + 2\pi k \qquad (k \text{ any integer})

are all coterminal with $\theta$ .

Worked example: Finding coterminal angles

Find the angle between $0$ and $2\pi$ that is coterminal with each angle, and name its quadrant.

$-\dfrac{\pi}{4}$
$\dfrac{11\pi}{3}$

Solutions.

Add one full turn: $-\dfrac{\pi}{4} + 2\pi = -\dfrac{\pi}{4} + \dfrac{8\pi}{4} = \dfrac{7\pi}{4}$ . Since $\dfrac{3\pi}{2} = \dfrac{6\pi}{4}$ and $2\pi = \dfrac{8\pi}{4}$ , the angle lies in Quadrant IV.
Subtract one full turn: $\dfrac{11\pi}{3} - \dfrac{6\pi}{3} = \dfrac{5\pi}{3}$ . That is $300^\circ$ , which is in Quadrant IV.

Arc length

Radians make arc length simple. An angle of $1$ radian cuts off an arc of length $r$ , so an angle of $\theta$ radians cuts off an arc $\theta$ times as long.

Arc length

On a circle of radius $r$ , a central angle of $\theta$ radians cuts off an arc of length

s = r\theta.

Common mistake

The formula $s = r\theta$ only works when $\theta$ is in radians. If you plug in degrees, the answer is off by a factor of about $57$ . Convert first: an angle of $60^\circ$ on a circle of radius $6$ gives $s = 6 \cdot \dfrac{\pi}{3} = 2\pi$ , not $6 \cdot 60 = 360$ .

Worked example: A swinging pendulum

A pendulum $80$ cm long swings through an angle of $45^\circ$ . How far does the tip travel?

Convert to radians: $45^\circ = \dfrac{\pi}{4}$ . The tip moves along an arc of a circle with radius $80$ cm:

s = r\theta = 80 \cdot \frac{\pi}{4} = 20\pi \approx 62.8 \text{ cm}.

The formula can also be solved for the angle: $\theta = \dfrac{s}{r}$ . This is exactly the definition of a radian in action: the angle counts how many radius-lengths fit along the arc.