Mathematics · Ch 3 — Trigonometric Functions

Radian and Degree Measure

There are two systems commonly used to measure the size of an angle.

1. Sexagesimal (degree) measure. A complete revolution is divided into $360$ equal parts,

each called one degree, written $1^\circ$ . Each degree is further divided into $60$ equal

parts called minutes ( $1^\circ = 60'$ ), and each minute into $60$ equal parts called

seconds ( $1' = 60''$ ). This system is inherited from ancient Babylonian astronomy and is

the one used in everyday and surveying contexts.

2. Circular (radian) measure. Let a circle of radius $r$ have an arc of length $s$ cut off

by an angle $\theta$ (in standard position) at its centre. The radian measure of $\theta$

is defined as the ratio

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

In particular, one radian is the angle subtended at the centre of a circle by an arc whose

length exactly equals the radius. Since this ratio of two lengths is a pure number, radian

measure has no physical unit attached to it (it is common to write "rad", but this is only a

reminder of the system used, not a unit of length).

Relating the two systems. For a complete revolution, $s$ equals the circumference $2\pi r$ ,

so the radian measure of one full revolution is $\dfrac{2\pi r}{r}=2\pi$ . Since one full

revolution also equals $360^\circ$ by definition,

2\pi \text{ radians} = 360^\circ \quad\Longrightarrow\quad \pi \text{ radians} = 180^\circ.

This single equation is the bridge between the two systems, and every conversion formula below

follows from it directly.

Converting degrees to radians. From $\pi$ radians $=180^\circ$ , dividing both sides by

$180$ gives $1^\circ = \dfrac{\pi}{180}$ radians. So to convert an angle from degree measure to

radian measure, multiply by $\dfrac{\pi}{180}$ :

\theta_{\text{radians}} = \theta_{\text{degrees}}\times\frac{\pi}{180}.

Converting radians to degrees. Equally, dividing $\pi$ radians $=180^\circ$ by $\pi$ gives

$1$ radian $=\dfrac{180}{\pi}$ degrees $\approx 57.296^\circ$ . So to convert an angle from

radian measure to degree measure, multiply by $\dfrac{180}{\pi}$ :

\theta_{\text{degrees}} = \theta_{\text{radians}}\times\frac{180}{\pi}.

Standard angles. The table below lists the radian measure of the angles used constantly

throughout this chapter.

Degree	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
Radian	$0$	$\pi/6$	$\pi/4$	$\pi/3$	$\pi/2$	$\pi$	$3\pi/2$	$2\pi$

Since $\pi$ is irrational, the radian measure of an angle that is a "nice" number of degrees …