I.
Introduction
Angles and Arcs
Angle – (def. geometry) the union of two rays that have common
endpoint.
Definition of an Angle
An angle is formed by rotating a given ray about its endpoint to some
terminal position. The original ray is the initial side of the angle, and the
second ray is the terminal side of the angle. The common endpoint is
the vertex of the angle.
Angles formed by a counterclockwise rotation are considered positive
angles, and angles formed by a clockwise rotation are considered
negative angles.
Angle Measurement
The measure of an angle is determined by the amount of rotation of
the initial side.
 360° - complete revolution
Classification of Angles According to their Measure
An angle superimposed in a Cartesian coordinate system is in
standard position if its vertex is at the origin and its initial side is on the
positive x-axis.
Definition of Degree
One degree is the measure of an angle formed by rotating a ray
of a complete revolution. The symbol for degree is °.
Protractor – used to measure the degree of an angle.
Two positive angles are complementary angles if the sum of the
measures of the angles is 90°. Each angle is the complement of the
other angle. Two positive angles are supplementary angles if the sum
of the measures of the angles is 180°. Each angle is the supplement of
the other angle.
Definition of Radian
One radian is the measure of the central angle subtended by an arc
of length r on a circle of radius r.
Definition of Radian Measure
Given an arc of length s on a circle of radius r, the measure of the
central angle subtended by the arc
radians.
If the terminal side of an angle in standard position lies on a
coordinate axis, then the angle is classified as a quadrantal angle.
Angles in standard position that have the same sides are coterminal
angles.
Radian Degree Conversion
To convert from radians to degrees, multiply by
To convert from degrees to radians, multiply by
Example:
1. Convert in degrees to radians:
a. 60°
b. 315°
c. -150°
2. Convert from radians to degrees:
a.
DMS Method (Decimal, Minute, Second)
a degree is subdivided into 60 equal parts, each of which is called a
minute, denoted by „. Thus 1° = 60‟. Furthermore, a minute is
subdivided into 60 equal parts, each of which is called a second,
denoted by “.
Decimal Degree Method
The measure 29.76° is a decimal degree. It means 29 plus 76
hundredths 1°.
Example Given: Convert 126°12‟27” in decimal degree.
radians
b.
radians
c.
radians
Arc Length Formula
Let r be the length of the radius of a circle and be the nonnegative
radian measure of a central angle of the circle. Then the length of the
arc s that subtends the central angle is
.
Example Given: Find the length of the arc that subtends a central
angle of 120° in a circle with a radius of 10cm.
Solve an Application:
A pulley with a radius of 10 inches uses a belt to drive a pulley with a
radius of 6 inches. Find the angle through which the smaller pulley
turns as the 10-inch pulley makes one revolution. State your answer in
radians and in degrees. (Hint s1 = s2)
Exercise set 1.
1. Find the measure (if possible) of the complement and supplement
of each angle.
a. 15 °
e. 56°33‟15‟‟
b. 22°43‟
f. 19°42‟05”
c.
g.
d.
h. 0.5 radian
2. Assume that the given angles are in standard position. Determine
the measure of the positive angle with measure less than 360
degrees that is coterminal with the given angle and then classify
the angle by quadrant.
a. 610°
c. 765°
b. -975°
d. -872°
3. Use calculator to convert each decimal degree to its equivalent
DMS measure.
a. 24.56°
c. 224.282°
b. 18.96°
d. 3.402°
4. Use calculator to convert each DMS measure to its equivalent
decimal degree measure.
a. 25°25‟12”
c. 19°12‟18”
b. 141°6‟9”
d. 183°33‟36”
5. Convert degree measure to exact radian measure.
6. Convert radian measure to degree measure.
7. Determine the arc length given the following dimensions: