Chapter 5: The Trigonometric
Functions
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Section 5-1: Angles and Degree
Measure
Objectives
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• Convert decimal degree measures to degree,
minutes, and seconds and vice versa
• Find the number of degrees in a given
number of rotations
• Identify angles that are co-terminal
with a given angle
History of Mathematics
•
Babylonians (4000-3000 B.C.) were some of the first peoples to leave samples of their
use of geometry in the form of inscribed clay tablets.
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•
The first written mathematical work containing definitions for angles was Euclid’s The
Elements. Little is known about the life of Euclid (about 300 B.C.), but his thirteenvolume work, The Elements, has strongly influenced the teaching of geometry for over
2000 years.
•
Euclid’s definition of a plane angle differed from an earlier Greek idea that an angle was
a deflection or a breaking of lines.
•
Greek mathematicians were not the only scholars interested in angles. Aristotle (384322 B. C.) had devised three categories in which to place mathematical concepts –a
quantity, a quality or a relation. Proclus (410-485) felt that an angle was a combination
of the three.
•
•
•
In 1634, Pierre Herigone first used “<“ as a symbol for an angle.
Today artists like Native American Autumn Borts use angles in their creation of pottery.
Angles
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• An angle may be generated by rotating one of two
rays that share a fixed endpoint
known as the vertex. One of the rays
is fixed to form the initial side and
the second ray rotates to form the
terminal side.
Terminal
α
side
Vertex
Initial side
Standard position
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• An angle with its vertex at the origin and its
initial side along the positive x axis is said
to be in standard position.
• The most common unit used to measure
angles is the degree.
Angle Measurement
•
The division of a circle into 360 degrees is based upon a unit of distance
by 1101
the Babylonians,
0011 0010used
1010
0001 0100 which
1011 was equal to 7 miles. They related time
and miles by observing the time it took for a person to travel one of their
units of distance. An entire day was 12 “time-miles”, so one complete
revolution of the sky by the sun was divided into 12 units. They also
divided each time-mile into 30 units. So 12 x 30 or 360 units represented
a complete revolution of the sun. Other historians feel that the
Babylonians used 360 degree in a circle because they had a base 60
number system and thought there were 360 days in a year.
• The Greeks adopted the 360 degree circle. Ptolemy (85-165) used the
degree system in his work in astronomy, including symbols for degree,
minutes and seconds in his mathematical work.
• The degree symbol o became more widely used after the publication of a
book by Dutch mathematician Gemma Frisius during the Renaissance in
1569.
• Radian angle measure was introduced by James Thomson in 1873. The
radian was developed to simplify formulas used in trigonometry and
calculus.
Quadrantal Angle
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• If the terminal side of an angle that is in
standard position coincides with one of the
axes, the angle is called a quadrantal
angle.
Co-terminal Angles
• Two angles in standard position are called coterminal angles if they have the same terminal side.
Since angles differing in degree measure by
multiples of 360 degrees are equivalent, every angle
has infinitely many co-terminal angles.
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If α is the degree measure of an angle, then
All angles measuring α + 360 k o , where k is
an integer, are co-terminal with α
Example # 1
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•
Give the angle measure represented by each rotation.
#1. 9.5 rotations clockwise
9.5 x 360 degrees = 3420 degrees
And since it is clockwise it is
-3420 degrees.
#2. 6.75 rotations counterclockwise
6.75 x 360 degrees = 2430
And since it is counterclockwise it is positive.
Example # 2
•
Identify all angles that are co-terminal with the angle. Then find one positive angle
one negative
angle0100
that are
co-terminal with the angle.
0011 0010and
1010
1101 0001
1011
86 degrees
To show all angles it would be 86 o + 360 k o
An example of a positive one would be
86 o + 360 (2) o
= 806 o
An example of a negative one would be
86o + 360(−1)o
= −274o
Example #3
•
If the angle is in standard position, determine a co-terminal angle that is between 0
degrees and 360 degrees. State the quadrant in which the terminal side lies.
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•
595 degrees
x + 360 (k )o = 595 o
k ≠ 2 because that would be 720 o
and k must be an integer, so k = 1
x + 360 o = 595o
x = 235o and it is in the thrid quadrant.
•
-777 degrees
x + 360(k ) o = −777 o
we know k must be negative
k = -2 then we have - 720o
which would mean x = -57o
− 57 o = 303o and that is in the fourth
quadrant.
Reference angle
• 0010
A reference
angle0001
is defined
the acute angle formed by the terminal
0011
1010 1101
0100as
1011
side of the given angle and the x axis.
Reference
angle
Reference
angle
Reference
Angle
Example
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• Find the measure of the reference angle for
the angle.
312 degrees
• #1 312 degrees
• ?= 360-312=48
• Reference angle is 48 degrees
?
Example
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• Find the reference angle for -195 degrees.
195-180-15
The reference angle is 15
degrees
?
-195 degrees
Hw#32
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•
•
•
•
Section 5-1
pp. 281-283
#18-26 all, 31-49 odds,
68,71,75,77