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Angles and Their Measure Objective: To define the measure of an angle and to relate radians and...
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Transcript of Angles and Their Measure Objective: To define the measure of an angle and to relate radians and...
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
000
000
32036040
40036040
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
000
000
240360120
480360120
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
b) 1200
c) 5200
• When an angle is greater than 3600, you should subtract 3600 twice rather than add it and subtract it.
000
000
200360160
160360520
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
• 135o
• -120o
000
000
33036030
30360390
000
000
225360135
495360135
000
000
480360120
240360120
Angles
• There are five different kinds of angles that we talk about.
Angle Pairs
• Two of the most talked about angle pairs are complimentary 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
000
000
13347180
434790
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 000 55125180
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.
= 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 = s/r, we can say that s = r.
Radians and Degrees
• Using the formula = s/r, we can say that s = r.• Since the circumference of a circle is r, we can say
the r = r.
Radians and Degrees
• Using the formula = s/r, we can say that s = r.• Since the circumference of a circle is r, we can say
the r = r.• Dividing each side by r, we get 2 = . This means
that the entire way around a circle is 2, so we know that 3600 = 2 radians.
Conversions
• Since 360o = 2 radians, we can say that 1800 = radians, or
• To convert a radian measure to degrees, multiply by
• Degrees to radians, multiply by
180
radian1180
reedeg1180
180
Conversions
• Convert to degrees. 2
Conversions
• Convert to degrees. 2
090180
2
Conversions
• Convert to degrees.
• Convert 1350 to radians.
2
090180
2
Conversions
• Convert to degrees.
• Convert 1350 to radians.
2
090180
2
4
3
180
135
1801350
You Try
• Convert to degrees.
• Convert 2100 to radians.
4
5
You Try
• Convert to degrees.
• Convert 2100 to radians.
4
5
0225180
4
5
6
7
180
210
180210
00
Arc Length
• In radians, arc length is easy. We use the equation
rs
Arc Length
• In radians, arc length is easy. We use the equation
• Find the arc length of a circle of radius 4 with a central angle of 3.
rs
Arc Length
• In radians, arc length is easy. We use the equation
• Find the arc length of a circle of radius 4 with a
central angle of 3.
rs
rs
1234 s
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:
rs 2
360
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:
• Find the arc length of a circle with an angle of 360 and a radius of 5.
rs 2
360
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:
• Find the arc length of a circle with an angle of 360 and a radius of 5.
rs 2
360
10
1052
360
36s
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 2.
Coterminal in Radians
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
6
Coterminal in Radians
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
6
2
6
Coterminal in Radians
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
6
2
6
6
12
6
6
136
11
Coterminal in Radians
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
6
2
6
6
12
6
6
136
11
4
3
Coterminal in Radians
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
6
2
6
6
12
6
6
136
11
4
34
8
4
3
4
114
5
You Try
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
3
7
You Try
• When working in degrees, to find coterminal angles, we added or subtracted 3600. In radians, we will add or subtract 2.
• Find one positive and one negative angle that is coterminal with:
3
733
6
3
7
3
5
3
6
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.
22
1rA
Sector of a Circle
• In radians, it is easy. Again we will just use an equation.
• Find the area of a sector of a circle with radius 6 and a central angle 3.
22
1rA
Sector of a Circle
• In radians, it is easy. Again we will just use an equation.
• Find the area of a sector of a circle with radius 6 and a central angle 3.
22
1rA
543)6(2
1 2 A
Degrees
• Again, degrees is a little bit harder. We are looking for part of the area of the circle. Since area is
the equation is:
2r
2
360rA
Degrees
• Again, degrees is a little bit harder. We are looking for part of the area of the circle. Since area is
the equation is:
• Find the area of a sector of a circle with radius 6 and central angle 900.
2r
2
360rA
Degrees
• Again, degrees is a little bit harder. We are looking for part of the area of the circle. Since area is
the equation is:
• Find the area of a sector of a circle with radius 6 and central angle 900.
2r
2
360rA
9)6(360
90 2 A
Homework
• Pages456-458• 5, 9, 11, 13, 15, 25, 31, 39, 41, 45, 47, 49, 51,
65, 67, 73-79 odd