Angles and Their Measure
Objective: To define the measure of
an angle and to relate radians and
degrees
Trigonometry
• In the Greek language, the word trigonometry means
“measurement of triangles.” Initially, trig dealt with
the relationships among the sides and angles of
triangles and was used in the development of
astronomy, navigation, and surveying. Now, it is
viewed more as the relationships of functions.
Angles
• An angle is determined by rotating a ray about its
endpoint. The starting position of the ray is called
the initial side of the angle, and the position after
rotation is the terminal side. The endpoint of the ray
is called the vertex of the angle. When the initial side
is the positive x-axis, it is in standard position.
Angles
• Positive angles are generated by counterclockwise
rotations starting at the positive x-axis.
• Negative angles are generated by clockwise rotations
starting at the positive x-axis.
Angles
• Positive angles are generated by counterclockwise
rotations.
• Negative angles are generated by clockwise
rotations.
• Angles are labeled with Greek letters or by using
three uppercase letters.
Coterminal Angles
• Angles that have the same initial side and terminal
side are called coterminal angles.
• There are an infinite number of angles that can be
coterminal.
Degree Measure
• The measure of an angle is determined by the
amount of rotation from the initial side to the
terminal side. The most common unit of angle
measure is the degree, denoted by the symbol 0.
Example 1
• Find two angles, one positive and one negative that
are coterminal with the following angles.
a) 400
Example 1
• Find two angles, one positive and one negative that
are coterminal with the following angles.
a) 400
400  3600  400 0
400  3600  320 0
Example 1
• Find two angles, one positive and one negative that
are coterminal with the following angles.
a) 400
b) 1200
Example 1
• Find two angles, one positive and one negative that
are coterminal with the following angles.
a) 400
b) 1200
1200  3600  4800
1200  3600  2400
Example 1
• Find two angles, one positive and one negative that
are coterminal with the following angles.
a) 400
b) 1200
c) 5200
Example 1
• Find two angles, one positive and one negative that
are coterminal with the following angles.
a) 400
5200  3600  1600
b) 1200
0
0
0
160  360  200
0
c) 520
•
When an angle is greater than 3600, you should
subtract 3600 twice rather than add it and subtract
it.
Example 1
• You Try:
• Find two angles, one positive and one negative that
are coterminal with the following angles.
• 390o
• 135o
• -120o
Example 1
• You Try:
• Find two angles, one positive and one negative that
are coterminal with the following angles.
• 390o
3900  3600  300
• 135o
300  3600  3300
• -120o
1350  3600  4950
1350  3600  2250
 1200  3600  2400
 1200  3600  4800
Angles
• There are five different kinds of angles that we talk
about.
Angle Pairs
• Two of the most talked about angle pairs are
complementary and supplementary angles.
• Complementary- Two angles whose sum is 900.
• Supplementary- Two angles whose sum is 1800.
Example 2
• If possible, find the complement and supplement for
the following angles.
a) 470
Example 2
• If possible, find the complement and supplement for
the following angles.
a) 470
900  47 0  430
1800  47 0  1330
Example 2
• If possible, find the complement and supplement for
the following angles.
a) 470
b) 1250
Example 2
• If possible, find the complement and supplement for
the following angles.
a) 470
b) 1250
1800  1250  550
Radians
• There is another way to express the measure of an
angle. This is called radians. To define a radian, you
can use a central angle (vertex at the center) of a
circle. The measure of the angle is the relationship
between the arc formed and the radius of the circle.
q = s/r
• A radian is the angle formed
• when the length of the arc (s)
• is equal to the radius of the circle (r).
Radians and Degrees
• Using the formula q = s/r, we can say that
s = rq.
Radians and Degrees
• Using the formula q = s/r, we can say that
s = rq.
• Since the circumference of a circle is 2pr, we can say
the 2pr = rq.
Radians and Degrees
• Using the formula q = s/r, we can say that
s = rq.
• Since the circumference of a circle is 2pr, we can say
the 2pr = rq.
• Dividing each side by r, we get 2p = q. This means
that the entire way around a circle is 2p, so we know
that 3600 = 2p radians.
Conversions
• Since 360o = 2p radians, we can say that
1800 = p radians, or
180
p
 1radian
p
180
 1 deg ree
• To convert a radian measure to degrees, multiply by
180
p
• Degrees to radians, multiply by
p
180
Conversions
p
• Convert
to degrees.
2
Conversions
p
• Convert
to degrees.
2
p 180

 900
2 p
Conversions
p
• Convert
to degrees.
2
p 180

 900
2 p
• Convert 1350 to radians.
Conversions
p
• Convert
to degrees.
2
p 180

 900
2 p
• Convert 1350 to radians.
135p 3p
135 


180 180
4
0
p
You Try
• Convert
5p
4
to degrees.
• Convert 2100 to radians.
You Try
• Convert
5p
4
to degrees.
5p 180

 2250
4 p
• Convert 2100 to radians.
p
0
210
p 7p
0
210 


180
180
6
Arc Length
• In radians, arc length is easy. We use the equation
s  rq
Arc Length
• In radians, arc length is easy. We use the equation
s  rq
• Find the arc length of a circle of radius 4 with a
central angle of 3.
Arc Length
• In radians, arc length is easy. We use the equation
s  rq
• Find the arc length of a circle of radius 4 with a
central angle of 3.
s  rq
s  4  3  12
Arc Length
• In degrees, it is a little bit harder. The entire way
around the circle is the circumference. We want part
of the circumference. The angle represents the part.
In degrees, arc length is:
s
q
360
 2pr
Arc Length
• In degrees, it is a little bit harder. The entire way
around the circle is the circumference. We want part
of the circumference. The angle represents the part.
In degrees, arc length is:
s
q
360
 2pr
• Find the arc length of a circle with an angle of 360
and a radius of 5.
Arc Length
• In degrees, it is a little bit harder. The entire way
around the circle is the circumference. We want part
of the circumference. The angle represents the part.
In degrees, arc length is:
s
q
360
 2pr
• Find the arc length of a circle with an angle of 360
and a radius of 5.
36
10p
s
 2p 5 
p
360
10
Class work
• Page 456
• 8a, 10, 14, 26, 46, 52, 66, 74, 76
Coterminal in Radians
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
Coterminal in Radians
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
p
6
Coterminal in Radians
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
p
p
 2p
6
6
Coterminal in Radians
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
p
p
p 12p
13p
11p
 2p


6
6
6
6
6
6
Coterminal in Radians
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
p
p
p 12p
13p
11p
 2p


6
6
6
6
6
6
3p
4
Coterminal in Radians
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
p
p
p 12p
13p
11p
 2p


6
6
6
6
6
6
3p
4
3p 8p

4
4
11p
4
5p

4
You Try
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
7p
3
You Try
• When working in degrees, to find coterminal angles,
we added or subtracted 3600. In radians, we will add
or subtract 2p.
• Find one positive and one negative angle that is
coterminal with:
7p
3
7p 6p p


3
3
3
p
6p
5p


3 3
3
Quadrants
• We will use the x-y coordinate graph to make 4
separate areas called quadrants. They are labeled
with roman numerals and go counterclockwise.
Sector of a Circle
• A sector of a circle is the region bounded by the two
radii of the circle and their intercepted arc.
Sector of a Circle
• In radians, it is easy. Again we will just use an
equation.
1 2
A r q
2
Sector of a Circle
• In radians, it is easy. Again we will just use an
equation.
1 2
A r q
2
• Find the area of a sector of a circle with radius 6 and
a central angle 3.
Sector of a Circle
• In radians, it is easy. Again we will just use an
equation.
1 2
A r q
2
• Find the area of a sector of a circle with radius 6 and
a central angle 3.
1 2
A  (6) 3  54
2
Degrees
• Again, degrees is a little bit harder. We are looking
2
p
r
for part of the area of the circle. Since area is
the equation is:
q
2
A
360
pr
Degrees
• Again, degrees is a little bit harder. We are looking
2
p
r
for part of the area of the circle. Since area is
the equation is:
q
2
A
360
pr
• Find the area of a sector of a circle with radius 6 and
central angle 900.
Degrees
• Again, degrees is a little bit harder. We are looking
2
p
r
for part of the area of the circle. Since area is
the equation is:
q
2
A
360
pr
• Find the area of a sector of a circle with radius 6 and
central angle 900.
90
A
p (6) 2  9p
360
Homework
• Pages456-458
• 5, 9, 11, 13, 15, 25, 31, 39, 41, 45, 47, 49, 51,
65, 67, 73-79 odd