14 angle

Some simple angles

360

π

2

Mathematicians sometimes find it convenient to deem

an angle measured in a counterclockwise sense as posi-

tive, and one measured in a clockwise sense as negative.

Babylonian scholars of ancient times were aware

that the year is composed of 365 days but chose to

operate with a convenient calendar composed of 12

months of 30 days. Thus the number 360 came to be

associated with the notion of a full cycle. Today, hark-

ing back to the Babylonians, angles are measured in

units of degrees, in which a full rotation corresponds to

360 degrees (written 360°). Thus a half turn corre-

sponds to 180°, and a quarter turn to 90°. A single

degree corresponds to 1/360 of a turn.

Each degree is divided into 60 smaller units called

minutes, denoted with an apostrophe, and each

minute is divided into 60 smaller units called seconds,

denoted with two apostrophes. Thus, for instance,

15°46′23′′ represents an angle of 15 degrees, 46 min-

utes, and 23 seconds.

Mathematicians prefer to use a unit of angle mea-

surement independent of the circumstance that we live

on the Earth, i.e., one that is natural to mathematics.

The chosen unit of measurement is called a radian.

Working with the simplest

CIRCLE

possible, namely, a

circle of radius one, mathematicians match the measure

of a full turn with the distance around that circle,

namely 2π, the circumference of the circle. Thus one

full rotation equals 2πradian. A half turn is measured

as half of this, namely, πradian, and a quarter turn as

radian.

To convert between degree and radian measures,

one simply notes that 360 degrees corresponds to 2π

radian. (Thus one degree equals radian,

and one radian equals .)

A unit of measurement called a “gradian” is some-

times used in surveying. In this system, a full turn is

considered 400 gradian (and, consequently, a quarter

turn is divided into exactly 100 parts). This system is

rarely used in mathematics, if at all.

Angles are classified according to their measure:

• An angle of zero degree is called a null angle.

• An angle between 0°and 90°is called acute.

• An angle of 90°is called a right angle. (It is the angle

formed when one makes a perfect right turn.)

• An angle between 90°and 180°is called obtuse.

• An angle of 180°is called a straight angle.

• An angle between 180°and 360°is called a reflex

angle.

• An angle of 360°is called a

PERIGON

or a round angle.

Two angles are said to be congruent if they have the

same measure. If two angles have measures summing to

a right angle, then they are said to be complementary,

and two angles are supplementary if their measures sum

to a straight angle. Special names are also given to angles

that appear in a diagram involving a

TRANSVERSAL

.

The angle of elevation of a point Pabove the

ground relative to an observer at position Qon the

ground is defined to be the angle between the line con-

necting Qto P, and the line connecting Qto the point

on the ground directly below P. If Plies below ground

level, then an angle of depression is computed similarly.

The angle between two

VECTOR

s is computed via

the

DOT PRODUCT

. Using

TRIGONOMETRY

one shows

that the angle Abetween two lines in the plane of

slopes m1and m2, respectively, is given by:

(This follows by noting that the angle the first line makes

with the x-axis is tan–1(m1) and the angle of the second

line is tan–1(m2). Thus the angle we seek is A= tan–1(m1)

– tan–1(m2). The trigonometric identity tan(α– β) =

now establishes the result.)

Consequently, if m1m2= –1, the lines are

PERPENDI

-

CULAR

. The angle between two intersecting curves in

a plane is defined to be the angle between the

TAN

-

GENT

lines to the curves at the point of intersection.

The link between the measure of an angle and the

length of arcs of a unit circle to define radian measure

can be extended to associate a measure of “angle” with