Download MTH 112, Class Notes, Name: , Date: Section 1.1, Angles 1. : a line

MTH 112, Class Notes, Name:
, Date:
Section 1.1, Angles
1.
: a line determined by two distinct points A and B.
Example.
2.
: the portion of the line AB between A and B including A
and B.
Example.
3.
: the portion of the line AB that starts at A and continues
through B, and on past B.
Example.
4.
: consists of two rays (or segments) in a plane with a common
endpoint.
Example.
5.
: the two rays (or segments) of an angle
6.
: the common endpoint
7. Associated with an angle is its
, generated by a rotation about its
This measure is determined by rotating a ray starting at one side of the angle, called the
, to the position of the other side, called the
.
Example.
8. Note. A
rotation gives a
Example.
rotation gives a
measure.
9. Note. An angle can be named by using
with the vertex letter
OR
or
.
angle
measure, while a
OR
. For example, the angle above can be called
1
.
,
2
MTH 112, Class Notes, Section 1.1, Angles
Degree Measure:
10.
: the most common unit for measuring angles.
Note. Degree measure was developed by the Babylonians 4000 years ago.
11. To use degree measure, we assign
Example.
degrees to a complete rotation of a ray.
Note. The Babylonians were the first to divide the circumference of a circle into 360 parts.
There are various theories as to why the number 360 was chosen. One is that it is approximately the number of days in a year, and it has many divisors, which makes it convenient
to work with.
Notice that the
when it makes a complete rotation.
of the angle corresponds to its
12.
One degree,
of a complete rotation.
represents
of a complete rotation.
represents
of a complete rotation.
: an angle measuring more than 0◦ but less than 90◦ .
13.
14.
, represents
: an angle measuring exactly 90◦ . The symbol q is often used
at the vertex of a right angle to denote the 90◦ measure.
15.
: an angle measuring more than 90◦ but less than 180◦ .
16.
: an angle measuring exactly 180◦ .
17. We often use the
angles.
Example.
to name
3
MTH 112, Class Notes, Section 1.1, Angles
18. If the sum of the measures of two positive angles is
and the angles are
, the angles are called
of each other.
19. If the sum of the measures of two positive angles is
and the angles are
, the angles are called
of each other.
Example 1. For an angle measuring 55◦ , find the measure of its (a) complement and (b)
supplement.
Example 2. Find the measure of each marked angle.
(a)
(b)
20. The measure of an angle A is often expressed by
angle A is 35◦ . Then we can say
21. Portions of a degree are measured in
:
of a degree
:
of a minute
. Suppose the measure of
.
and
.
4
MTH 112, Class Notes, Section 1.1, Angles
Example 3.
Perform each calculation:
a. 28◦ 35′ + 63◦ 52′
22. Angles can also be measured in
Example.
b. 180◦ − 117◦ 29′
.
Example 4.
a. Convert 105◦ 20′ 32′′ to decimal degrees to the nearest thousandth.
b. Convert 85.263◦ to degrees, minutes, and seconds.
5
MTH 112, Class Notes, Section 1.1, Angles
Standard Position:
23. An angle is in
initial side lies on the
Example.
if its vertex is at the
and its
x-axis.
24. An angle in standard position is said to
lies. An
an
angle is in quadrant
Example.
in the quadrant in which its
angle is in quadrant
and
.
25. Angles in standard position whose terminal sides lie on the x-axis or y-axis, such as angles
with measures
,
,
, and so on, are called
.
6
MTH 112, Class Notes, Section 1.1, Angles
Coterminal Angles:
26. A complete rotation of a ray results in an angle measuring
rotation, angles of measure larger than 360◦ can be produced.
Example.
27. These angles have the same
Such angles are called
.
. By continuing the
side, but different amounts of
.
; their angles differ by a multiple of
Example 5. Find the angles of least possible positive measure coterminal with each angle.
(a) 1106◦
(b) −150◦
(c) −603◦
28. Sometimes it is necessary to find an expression that will generate all angles coterminal
with a given angle. For example, we can obtain any angle coterminal with 60◦ by adding
an appropriate integer multiple of
to 60◦ . Let
represent any
;
then the expression
represents all such coterminal angles.
Example 6. A wheel makes 270 revolutions per minute. Through how many degrees will
a point on the edge of the wheel move in 5 seconds?
Homework: