Math Core

In geometry, an angle was a fixed shape between two rays, and it never measured more than $360^\circ$ . Trigonometry treats an angle as a rotation, which can be as large as you like and can turn either way. That view needs a second unit, the radian, which links angles directly to lengths along a circle and is the unit that calculus and physics use.

Angles as rotations

Picture a ray pinned at its endpoint. It starts in one position, the initial side, and rotates to a new position, the terminal side. The pinned endpoint is the vertex. The angle measures how far the ray turned.

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$ -axis. The direction of the turn matters:

A counterclockwise rotation gives a positive angle.
A clockwise rotation gives a negative angle.

The terminal side of an angle in standard position lands in one of the four quadrants, or on an axis. An angle whose terminal side lies on an axis (such as $90^\circ$ or $180^\circ$ ) is called a quadrantal angle.

An angle of 150° in standard position. The ray turns counterclockwise from the positive x-axis into Quadrant II.

Degrees

You already know the degree: one full rotation is $360^\circ$ , so $1^\circ$ is $\dfrac{1}{360}$ of a turn. A quarter turn is $90^\circ$ and a half turn is $180^\circ$ .

Degrees can be split further. One degree equals $60$ minutes ( $60'$ ), and one minute equals $60$ seconds ( $60''$ ). To change degrees-minutes-seconds to decimal degrees, divide the minutes by $60$ and the seconds by $3600$ :

24^\circ 15' = 24 + \frac{15}{60} = 24.25^\circ.

Radians

Draw a circle of radius $r$ centered at the vertex of an angle. The angle cuts off an arc of the circle. If that arc is exactly as long as the radius, the angle measures one radian.

A central angle of 1 radian cuts off an arc that is exactly one radius long.

Definition

Radian

A central angle has measure $1$ radian when the arc it intercepts is equal in length to the radius of the circle. In general, if an angle cuts off an arc of length $s$ on a circle of radius $r$ , its measure in radians is

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

Since $\theta = \dfrac{s}{r}$ is a length divided by a length, a radian has no physical units. That is why an angle like $\theta = 2$ with no unit written is understood to be $2$ radians.

How many radians are in a full turn? The whole circumference is $2\pi r$ , so a full rotation measures

\frac{2\pi r}{r} = 2\pi \text{ radians}.

Since $2\pi$ is a little more than $6$ , about six radius-length arcs fit around any circle, with a bit left over.

Converting between the units

A full turn is both $360^\circ$ and $2\pi$ radians. Dividing both by $2$ gives the one fact you need:

The conversion fact

180^\circ = \pi \text{ radians}

Degrees to radians: multiply by $\dfrac{\pi}{180^\circ}$ .
Radians to degrees: multiply by $\dfrac{180^\circ}{\pi}$ .

Each conversion factor equals $1$ , so multiplying by it changes the units without changing the angle. Pick the factor whose units cancel: degrees on the bottom to get rid of degrees, $\pi$ on the bottom to get rid of radians.

Worked example: Degrees to radians

Convert $150^\circ$ and $-225^\circ$ to radians. Give exact answers.

150^\circ \cdot \frac{\pi}{180^\circ} = \frac{150\pi}{180} = \frac{5\pi}{6}

-225^\circ \cdot \frac{\pi}{180^\circ} = -\frac{225\pi}{180} = -\frac{5\pi}{4}

Leave $\pi$ in the answer and reduce the fraction. The negative sign stays: $-\dfrac{5\pi}{4}$ is still a clockwise rotation.

Worked example: Radians to degrees

Convert $\dfrac{7\pi}{4}$ to degrees.

\frac{7\pi}{4} \cdot \frac{180^\circ}{\pi} = \frac{7 \cdot 180^\circ}{4} = 7 \cdot 45^\circ = 315^\circ

A quicker way: since $\pi$ means $180^\circ$ , replace $\pi$ by $180^\circ$ and simplify. $\dfrac{7(180^\circ)}{4} = 315^\circ$ .

Worked example: Radians without π

Convert $2.5$ radians to degrees, to the nearest tenth.

2.5 \cdot \frac{180^\circ}{\pi} = \frac{450^\circ}{\pi} \approx 143.2^\circ

So $2.5$ radians is a little less than $\pi \approx 3.14$ radians, which matches $143.2^\circ$ being a little less than $180^\circ$ . Its terminal side is in Quadrant II. As a rule of thumb, $1$ radian is about $57.3^\circ$ .

Common mistake

Not every radian measure contains $\pi$ . The angle $2$ means $2$ radians, which is about $114.6^\circ$ . It does not mean $2^\circ$ , and it is not $2\pi$ . Also check your calculator's mode: in degree mode, $\sin 2$ gives the sine of $2^\circ$ , which is a different number from the sine of $2$ radians.

Angles worth memorizing

These angles appear constantly. Learn them as fractions of $\pi$ , the way you already know fractions of a full turn.

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

Tip

Think in "slices of $\pi$ ." Since $\dfrac{\pi}{6} = 30^\circ$ , the angle $\dfrac{5\pi}{6}$ is five $30^\circ$ slices, or $150^\circ$ . Since $\dfrac{\pi}{4} = 45^\circ$ , the angle $\dfrac{7\pi}{4}$ is seven $45^\circ$ slices, or $315^\circ$ . This also tells you the quadrant at a glance: $\dfrac{5\pi}{6}$ is less than $\pi$ but more than $\dfrac{\pi}{2}$ , so it is in Quadrant II.