Angles and their measurement

Trigonom­e­try began as the mea­sure­ment of tri­an­gles, but it soon grew into the study of angles of any size: the turn of a wheel, the swing of a pen­du­lum, the rota­tion of the Earth. So we start by look­ing at an angle in a new way, as an amount of turn­ing, and by learn­ing the three sys­tems used to mea­sure it: degrees, grades and radi­ans. The radian is the one math­e­mati­cians use most, and con­vert­ing between radi­ans and degrees quickly is a skill you will need in every later les­son.

An angle as a rota­tion#

An angle is the amount of rota­tion of a revolv­ing ray about its end point, mea­sured from a fixed start­ing posi­tion. The fixed ray is the ini­tial side; the ray in its final posi­tion is the ter­mi­nal side.

If the rota­tion is anti­clock­wise, the angle is taken as pos­i­tive. If it is clock­wise, the angle is neg­a­tive.

Because a ray can keep turn­ing, angles are not lim­ited to $180°$ or $360°$: two full turns anti­clock­wise make $720°$, and a quar­ter turn clock­wise is $-90°$.

Sex­a­ges­i­mal sys­tem (degrees)#

In this sys­tem, used in Britain and in most of our work:

• $\displaystyle \frac{1}{360}$ of a com­plete rota­tion is one degree, writ­ten $1^\circ$;
• $\displaystyle \frac{1}{60}$ of a degree is one minute, writ­ten $1'$;
• $\displaystyle \frac{1}{60}$ of a minute is one sec­ond, writ­ten $1''$.

So $1^\circ = 60'$ and $1' = 60''$. A com­plete angle is $360^\circ$, a straight angle $\displaystyle \frac{360^\circ}{2} = 180^\circ$ and a right angle $\displaystyle \frac{360^\circ}{4} = 90^\circ$.

Cen­tes­i­mal sys­tem (grades)#

In this sys­tem, which comes from France:

• $\displaystyle \frac{1}{400}$ of a com­plete rota­tion is one grade, writ­ten $1^g$;
• $\displaystyle \frac{1}{100}$ of a grade is one cen­tes­i­mal minute;
• $\displaystyle \frac{1}{100}$ of a cen­tes­i­mal minute is one cen­tes­i­mal sec­ond.

A com­plete angle is $400^g$ and a right angle is $\displaystyle \frac{400^g}{4} = 100^g$. The min­utes and sec­onds of this sys­tem are dif­fer­ent from the min­utes and sec­onds of the sex­a­ges­i­mal sys­tem, even though they share the names.

Cir­cu­lar sys­tem (radi­ans)#

A radian is the angle sub­tended at the cen­tre of a cir­cle by an arc whose length equals the radius of the cir­cle. It is writ­ten $1^c$.

This def­i­n­i­tion does not depend on the size of the cir­cle. The whole cir­cum­fer­ence is $2\pi r$, which is $2\pi$ radius-lengths, so a com­plete angle is $2\pi$ radi­ans. When an angle is writ­ten with­out a unit, such as $\displaystyle \frac{\pi}{3}$, it means radi­ans.

Con­nect­ing the sys­tems#

The com­plete angle in each sys­tem gives the key link:

$$\displaystyle 360^\circ = 2\pi^c = 400^g, \qquad 180^\circ = \pi^c = 200^g, \qquad 90^\circ = \frac{\pi}{2} = 100^g$$

Mul­ti­ples of $360^\circ$ fol­low: $720^\circ = 4\pi$, $1080^\circ = 6\pi$, $1440^\circ = 8\pi$, $1800^\circ = 10\pi$. And of $90^\circ$: $\displaystyle 270^\circ = \frac{3\pi}{2}$, $\displaystyle 450^\circ = \frac{5\pi}{2}$, $540^\circ = 3\pi$, $\displaystyle 630^\circ = \frac{7\pi}{2}$.

To con­vert, use

$$\displaystyle \text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}$$

and for grades, $\displaystyle \frac{D}{90} = \frac{G}{100}$, where $D$ is the angle in degrees and $G$ in grades.

Angles worth know­ing in radi­ans#

Many oth­ers are built from these: $\displaystyle 210^\circ = 180^\circ + 30^\circ = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$, $\displaystyle \;225^\circ = \frac{5\pi}{4}$, $\displaystyle \;240^\circ = \frac{4\pi}{3}$, $\displaystyle \;300^\circ = 360^\circ - 60^\circ = \frac{5\pi}{3}$, $\displaystyle \;315^\circ = \frac{7\pi}{4}$, $\displaystyle \;330^\circ = \frac{11\pi}{6}$. Also $\displaystyle 15^\circ = \frac{\pi}{12}$, $\displaystyle 18^\circ = \frac{\pi}{10}$, $\displaystyle 36^\circ = \frac{\pi}{5}$, $\displaystyle 54^\circ = \frac{3\pi}{10}$, $\displaystyle 72^\circ = \frac{2\pi}{5}$, $\displaystyle 22\frac{1}{2}^\circ = \frac{\pi}{8}$ and $\displaystyle 67\frac{1}{2}^\circ = \frac{3\pi}{8}$.

How big is a radian?#

Using $\displaystyle \pi = \frac{22}{7}$:

$$\displaystyle 1^\circ = \frac{\pi}{180} = \frac{22}{7 \times 180} \approx 0.01746 \text{ radian}$$

$$\displaystyle 1^c = \frac{180^\circ}{\pi} = \frac{180 \times 7}{22} \approx 57.27^\circ \approx 57^\circ 16' 22''$$

So one radian is a lit­tle under $60^\circ$, much larger than one degree.

Length of an arc#

Since a radian is the angle for an arc of one radius, an angle of $\theta$ radi­ans cuts off an arc of $\theta$ radii:

$$l = r\theta \quad (\theta \text{ in radians})$$

Exam­ple 1#

Con­vert $75^\circ$ and $40^\circ 20'$ to radi­ans.

$$\displaystyle 75^\circ = 75 \times \frac{\pi}{180} = \frac{5\pi}{12}$$

$\displaystyle 40^\circ 20' = 40\frac{1}{3}^\circ = \frac{121}{3}^\circ$, so in radi­ans it is $\displaystyle \frac{121}{3} \times \frac{\pi}{180} = \frac{121\pi}{540}$.

Exam­ple 2#

Con­vert $\displaystyle \frac{7\pi}{6}$ to degrees, and $1.2$ radi­ans to degrees, min­utes and sec­onds (use $\displaystyle \pi = \frac{22}{7}$).

$$\displaystyle \frac{7\pi}{6} = \frac{7 \times 180^\circ}{6} = 210^\circ$$

$$\displaystyle 1.2^c = 1.2 \times \frac{180 \times 7}{22} = \frac{1512}{22} \approx 68.727^\circ$$

The dec­i­mal part gives $0.727 \times 60 \approx 43.64'$, and $0.64 \times 60 \approx 38''$. So $1.2^c \approx 68^\circ 43' 38''$.

Exam­ple 3#

Con­vert $60^\circ$ to grades, and $50^g$ to degrees.

$\displaystyle \frac{60}{90} = \frac{G}{100}$ gives $\displaystyle G = \frac{200}{3} = 66\frac{2}{3}$, so $\displaystyle 60^\circ = 66\frac{2}{3}^g$. And $\displaystyle \frac{D}{90} = \frac{50}{100}$ gives $D = 45$, so $50^g = 45^\circ$.

Exam­ple 4#

Find the length of an arc of a cir­cle of radius $14$ cm that sub­tends $45^\circ$ at the cen­tre. Also find the angle, in degrees, sub­tended by an arc of $33$ cm in a cir­cle of radius $21$ cm.

$\displaystyle 45^\circ = \frac{\pi}{4}$, so $\displaystyle l = 14 \times \frac{22}{7} \times \frac{1}{4} = 11$ cm.

$\displaystyle \theta = \frac{l}{r} = \frac{33}{21} = \frac{11}{7}$ radi­ans, which is $\displaystyle \frac{11}{7} \times \frac{180 \times 7}{22} = 90^\circ$.

Prac­tice#

1. Con­vert to radi­ans: (a) $30^\circ$ (b) $135^\circ$ (c) $240^\circ$
2. Con­vert to degrees: (a) $\displaystyle \frac{\pi}{9}$ (b) $\displaystyle \frac{5\pi}{4}$ (c) $\displaystyle \frac{11\pi}{6}$
3. Con­vert $22^\circ 30'$ to radi­ans.
4. Con­vert $3$ radi­ans to degrees, min­utes and sec­onds, using $\displaystyle \pi = \frac{22}{7}$.
5. Con­vert $45^\circ$ to grades and $120^g$ to degrees.
6. Find the length of an arc of a cir­cle of radius $7$ cm that sub­tends $60^\circ$ at the cen­tre.
7. A wheel of radius $35$ cm turns so that a point on its rim moves $55$ cm. Through what angle, in radi­ans and in degrees, has the wheel turned?
8. The angles of a tri­an­gle are in the ratio $1 : 2 : 3$. Find them in radi­ans.
9. Which is larger, $1^\circ$ or $1^c$? Give the approx­i­mate size of each in the other unit.

Answers#