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