The University of Memphis
Department of Mathematical Sciences
D. P. Dwiggins, PhD
Fall 2006

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Concepts of Trigonometry

Angles and Angular Measure


Recall the Euclidean postulate which states that two lines intersecting at a single point uniquely determine
a plane. Moreover, these two lines divide the plane into four distinct regions, referred to as angles.
Definition: An angle is a region in the plane which is partially bounded on two sides by two lines
intersecting at a single point. Note by this definition the angle itself consists of all the points in a particular
region of the plane. For example, in the Cartesian coordinate system (number plane), the four quadrantal
angles are referred to as Quadrants I, II, III, and IV.
The lines forming the boundaries of an angle are called the sides of the angle, and the point where the
lines intersect is called the vertex of the angle. Let A be the point of intersection (the vertex), and let
B and C be points on the two sides of the angle. Then the angle is referred to symbolically as (Angle)BAC.
It doesn't matter which of the two sides is listed first, so (Angle)CAB is the same as (Angle)BAC.
That is, the order of the points labeling the angle does not matter, as long as the vertex is listed in the middle.

Now, planes are infinite in extent (this is one of Euclid's "axioms"), which may present a problem in deciding
how angles should be measured. However, there is another approach using circles which may be used to
define angles as being bounded in extent. Let a circle C be given, centered at vertex A, and let P and Q be
the points where the circle intersects the sides of the angle. Then (Angle)PAQ (equivalently, (Angle)QAP)
is referenced within the circle as a central angle (since the center of the circle is the vertex of the angle).
In this setting, the region of the plane bounded by the radii AP and AQ and the arc PQ is called a sector
of the circle. (Note: an arc of a circle is defined as a portion of its circumference lying between two end
points, just as a line segment is a portion of a line lying between two end points.) Also in this setting, the
angle PAQ is said to be subtended by the arc PQ. Note that if a different circle C' is drawn centered at
A but with a different radius, then C' intersects the sides of the angle at different points P' and Q', but the
angle P'AQ' is supposed to be the same as the original angle PAQ, which again raises the question of how
angles are supposed to be measured.

Angular measure is a way to compare the relative "size" of two angles. Angular regions of the plane are
of infinite extent, so it is impossible to use planar measure (area) to define the size of an angle, and it is also
impossible to use circular arclength alone to define angular measure, as discussed in the paragraph above.
Angular sizes are not measured in terms of an "absolute" unit of angular measure, such as what is done for
length (the meter) or weight (the gram). Instead, angles are measured relative to each other, or relative to
the angle representing the entire plane (or, for central angles, the entire circle). It is perhaps easier to see
how central angles might be measured in terms of fractions of a circle, but let us return to the original
definition of an angle and see how it might be defined in terms of fractions of the plane.

Given two lines interesecting at a single point, defining four planar angles, use the Euclidean postulate of
measure so that both lines represent real number lines, intersecting at the common origin (0,0). Each number
line is marked off in terms of a unit of distance, which is marked off from (0,0) in both directions along both
the positive and negative directions of each number line, defining the four "compass points" (1,0), (-1,0),
(0,1), and (0,-1). Consider now the following four distances between these four points:

      • D1 = distance between (1,0) and (0,1) = the Quadrant I distance
      • D2 = distance between (-1,0) and (0,1) = the Quadrant II distance
      • D3 = distance between (-1,0) and (0,-1) = the Quadrant III distance
      • D4 = distance between (1,0) and (0,-1) = the Quadrant IV distance

If the number lines intersect to form angles in such a way that all four of these distances have the same
measure, then the lines are said to be perpendicular, and all four quadrantal angles are defined to have
the same measure. In this case each angle is said to be a right angle, and so a right angle has measure
representing one-fourth of the plane when the plane is divided into four equal parts. (Note: "perpendicular"
is based on a Latin phrase meaning "vertical" in the sense that a weight hanging from a string will cause the
string to be vertical as opposed to the horizontal ground ("horizontal" of course referring to the ground's
horizon), and if you want your building to stand up straight then the walls must form the "right" angle with
respect to the ground, i.e. erected to follow the same direction as the perpendicular line.)

If the two number lines interesect in such a way that the four distances D1 through D4 are not all the same,
then the angles formed by the lines will have different measure, either greater than or less than the measure
of a right angle. However, since it is completely arbitrary along each number line which direction is chosen
as positive and which direction is negative, it will always be true in the list above that D1 is equal to D3 and
D2 is equal to D4. Thus, if relative distances are to be used in defining angular measure, when two lines
intersect at a single point they must always form two pairs of angles such that within each pair of angles
the two have the same measure. Thus, two lines interesecting at a single point always form two pairs of
angles of the same measure, referred to as vertical angles, and if the lines are not perpendicular then in
one pair of vertical angles each angle has measure less than that of a right angle (acute angles) and in the
other pair of vertical angles each angle has measure greater than that of a right angle (obtuse angles).

Units of Angular Measure

Now that the idea of relative angular measure has been established, return to the concept of central angles
defined in terms of a circle. By definition, right angles must divide the circle into four equal parts, and so,
once it is decided what angular measure an entire circle has, a right angle must have angular measure
equal to that of a quarter-circle. There are two basic circular units of angular measure:
Degree Measure defines the circle as having measure 360 (approximately the number of days
in a year), and so in this system a right angle would have measure 90 degrees. This was the system
developed by early mathematicians, in part because 360 is divisible by many whole numbers.
The second system is that of Radial Measure, which measures the size of a central angle subtended
by a given arc in terms of the ratio of the arclength along the circumference of the circle to that of the radius
of the given circle. Radial measure, the units of which are called radians, must then be given in terms of the
number pi, which is defined to be the ratio of any circle's circumference to its diameter (this ratio is a little
bit bigger than 3). That is, C = pi*D, where C is the circumference of a circle and D is its diameter.
Since the radius (r) of a circle is twice its diameter, and since radians measure the ratio of circumferential
arclength to the radius, the entire circle must have radial measure equal to 2*pi radians. Thus, in radial
measure, a right angle (quarter-circle) has measure pi/2 ("pi-halves") radians.

Note that both these units of angular measure are defined by dividing a circle into a certain number of
parts, with the entire circle measured as either 360 degrees or 2*pi radians. Thus, one degree is equal
to pi/180 radians, and one radian is equal to 180/pi degrees. Since pi is a little bigger than 3, one radian
is a little bit less than 60 degrees.

Other units of angular measurement are based on dividing a right angle into a certain number of equal parts.
For example, gradial measure divides a right angle into 100 decimal parts, so that 90 degrees is the same
as 100 grads. Decimal angular measure is useful in engineering calculations, which is often applied to the
military, so there is also a finer subdivision of angular measure used in military applications referred to as a
mil, with different definitions depending upon the application, but generally if you bisect a grad and then
bisect it again you get a mil, which means there are 400 grads and 1600 mils in a circle

The number of degrees in a circle, 360, is a multiple of 2, 3, 4, 5, 6, (not 7, a magic number), 8, 9, 10,
(not 11, the next magic number), 12, (not 13 or 14), 15, 20, 30, 45, 90, and 180, so ancient mathematicans
found 360 to be useful in many calculations, especially in astronomy because 360 is very nearly the number
of days it takes the Earth to go around the Sun. Suppose life had developed on Mars instead of Earth?
How many degrees would there be in a Martian circle, a Jovian circle, or an Alpha Centaurin circle?

How would you define angular measure?