section 1.1 angle measure and arc length section 1.1 angle measure and arc length

Section 1.1Angle Measure and Arc

Length

Section 1.1Angle Measure and Arc

Length

Objectives:1. To draw an angle in standard

position.2. To convert between degree and

radian measure.3. To calculate arc length using

angle measure and radius length.4. To review the distance and

midpoint formulas.

Objectives:1. To draw an angle in standard

position.2. To convert between degree and

radian measure.3. To calculate arc length using

angle measure and radius length.4. To review the distance and

midpoint formulas.

An angle in standard position has the positive x-axis as its initial ray and the origin as its vertex.

An angle in standard position has the positive x-axis as its initial ray and the origin as its vertex.

DefinitionDefinitionDefinitionDefinition

Standard PositionStandard Position

initial rayinitial ray

terminal rayterminal ray

VV

CC

AA

BB CC

AA

6060

Standard PositionStandard Position

Standard PositionStandard Position

FF

GGEE -120-120

AA

CCBB

8080

Standard PositionStandard Position

Coterminal angles are angles in standard position with the same terminal ray.

Coterminal angles are angles in standard position with the same terminal ray.

DefinitionDefinitionDefinitionDefinition

Coterminal AnglesCoterminal Angles

AA

CCBB440440

AA

CCBB

8080

A radian is the measure of an angle formed by two radii of a circle so that the intercepted arc has a length equal to the radius of the circle.

A radian is the measure of an angle formed by two radii of a circle so that the intercepted arc has a length equal to the radius of the circle.

DefinitionDefinitionDefinitionDefinition

RadianRadian

BB CC

AA

mABC = 1 radian 57.3mABC = 1 radian 57.3

Whenever an angle measure is given without degrees specified as the unit of measure, it is measured in radians.

Whenever an angle measure is given without degrees specified as the unit of measure, it is measured in radians.

The unit circle has its center at the origin with a radius of one unit.

The unit circle has its center at the origin with a radius of one unit.

DefinitionDefinitionDefinitionDefinition

Unit CircleUnit Circle

BB CC

AA

(1, 0)(1, 0)

Convert between degrees and radians by setting up a proportion.

Convert between degrees and radians by setting up a proportion.

180180

ππ==

degdegmm

radradnn

In this proportion, substitute the known measure and solve for the other measure.

In this proportion, substitute the known measure and solve for the other measure.

Change 60 degrees to radians.Change 60 degrees to radians.

m degm degn radn rad

180180

==

Change 60 degrees to radians.Change 60 degrees to radians.

60 deg60 degn radn rad

180180

==

n(180) = 60n(180) = 60

n =n = 1.051.0533

Change 5.6 radians to degrees.Change 5.6 radians to degrees.

m degm degn radn rad

180180

==

Change 5.6 radians to degrees.Change 5.6 radians to degrees.

m degm deg5.6 rad5.6 rad

180180

==

5.6(180) = m5.6(180) = m

m =m = 320.9° 320.9°5.6(180)5.6(180)

EXAMPLE 1 Find the measure of in degrees and radians.EXAMPLE 1 Find the measure of in degrees and radians.

= ¼ of a circle

¼(360º) = 90º

¼ • 2 =

= ¼ of a circle

¼(360º) = 90º

¼ • 2 =22

Practice Question: Find the measure of in degrees.Practice Question: Find the measure of in degrees.

= ¾ of a circle

¾(360º) = 270º

= ¾ of a circle

¾(360º) = 270º

A quadrantal angle is an angle whose terminal ray lies on one of the axes.

A quadrantal angle is an angle whose terminal ray lies on one of the axes.

DefinitionDefinitionDefinitionDefinition

Quadrantal AngleQuadrantal Angle

BB CC

AA

90°90°

BB CC

AA

180°180°

Quadrantal AngleQuadrantal Angle

BB CC

AA

270°270°

Quadrantal AngleQuadrantal Angle

BB CC

AA

360°360°

Quadrantal AngleQuadrantal Angle

Concentric circles have the same center but different radii. A central angle is an angle having its vertex at the center of a circle.

Concentric circles have the same center but different radii. A central angle is an angle having its vertex at the center of a circle.

DefinitionDefinitionDefinitionDefinition

(r,0)(r,0)

(1,0)(1,0)

x

y

ss

tt

From geometry it can be proved that the ratio of the arc length of two concentric circles is the same as the ratio of their radii:

From geometry it can be proved that the ratio of the arc length of two concentric circles is the same as the ratio of their radii:

11rr

ttss

==

Solving for s, s = rt , where s = arc length, r = radius, and t = angle measure (in radians).

Solving for s, s = rt , where s = arc length, r = radius, and t = angle measure (in radians).

In other words, the length of any arc equals the product of its radius and angle measure (in radians)

In other words, the length of any arc equals the product of its radius and angle measure (in radians)

Find the measure of the angle that cuts off an arc length of 8 in a circle with radius 2.

Find the measure of the angle that cuts off an arc length of 8 in a circle with radius 2.

s = rt8 = 2(t)t = 4 radians

s = rt8 = 2(t)t = 4 radians

EXAMPLE 2 Find the measure of the angle that cuts off an arc length of 7 in a circle of radius 4.

EXAMPLE 2 Find the measure of the angle that cuts off an arc length of 7 in a circle of radius 4.

Answer s = rt

7 = 4t

t = 7/4 = 1.75 radians

Answer s = rt

7 = 4t

t = 7/4 = 1.75 radians

Practice Question: Find the measure of the angle that cuts off an arc length of 8 in a circle of radius 5.

Practice Question: Find the measure of the angle that cuts off an arc length of 8 in a circle of radius 5.

Answer s = rt

8 = 5t

t = 8/5 = 1.6 radians

Answer s = rt

8 = 5t

t = 8/5 = 1.6 radians

Homework:

pp. 6-8

Homework:

pp. 6-8

►A. ExercisesSketch each angle in standard position.

3. 2/3

►A. ExercisesSketch each angle in standard position.

3. 2/3

00

22

32

32

Sketch each angle in standard position.

290°

Sketch each angle in standard position.

290°

00180°180°

270°270°

►A. ExercisesComplete each line of the following table.►A. ExercisesComplete each line of the following table.

Radians Degrees

9. 15

11.

15.

17. 31

6

7

7

12

/12/12

210210

309309

0.540.54

►B. ExercisesGive three angle measures in radians which are coterminal with each of the following. Include at least one positive and one negative angle measure.

21.

►B. ExercisesGive three angle measures in radians which are coterminal with each of the following. Include at least one positive and one negative angle measure.

21.44

►B. Exercises23. How many radians are in one degree?►B. Exercises23. How many radians are in one degree?

n rad (180) = degn rad (180) = deg

1 deg1 degn radn rad

180180

==

n =n =180180

0.01745 0.01745

►B. ExercisesFind the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

►B. ExercisesFind the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

Arc Length Radius Length Angle Measure(s) (r) (t)

8

5

4

2

6

32

25.

27.

29.

25.

27.

29.

Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

25. Given r = 8 and t = /4, use s = rt to

find the arc length (s).

Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

25. Given r = 8 and t = /4, use s = rt to

find the arc length (s).s = rt

= 8(/4)

= 2

s = rt= 8(/4)

= 2

Arc Length Radius Length Angle Measure(s) (r) (t)

8

5

4

2

6

32

22

►B. ExercisesFind the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

►B. ExercisesFind the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

25.

27.

29.

25.

27.

29.

Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

27. Given s = /2 and t = /6, use s = rt to

find the radius length (r).

Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

27. Given s = /2 and t = /6, use s = rt to

find the radius length (r).s = rt

/2 = r(/6)

r = 6/(/2) = 3

s = rt/2 = r(/6)

r = 6/(/2) = 3

Arc Length Radius Length Angle Measure(s) (r) (t)

8

5

4

2

6

32

22

33

►B. ExercisesFind the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

►B. ExercisesFind the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians.

25.

27.

29.

25.

27.

29.

►B. ExercisesDetermine the distance between each pair of points and also the midpoint of the segment joining them.

30. (3, -6), (1, 0)

►B. ExercisesDetermine the distance between each pair of points and also the midpoint of the segment joining them.

30. (3, -6), (1, 0)

d = (x2 – x1)2 + (y2 – y1)2d = (x2 – x1)2 + (y2 – y1)2

midpoint = ( , )midpoint = ( , )x1 + x2

2x1 + x2

2y1 + y2

2y1 + y2

2

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

(9, -2) =(9, -2) =22

5 + y5 + y,,

223 + x3 + x

18 = 3 + x18 = 3 + x22

3 + x3 + x9 =9 =

x = 15x = 15

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

-4 = 5 + y-4 = 5 + y22

5 + y5 + y-2 =-2 =

y = -9y = -9

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

B = (15, -9)B = (15, -9)

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)

►B. ExercisesIf M is the midpoint of AB, find B. Then use distances to justify your answers.

35. A(3, 5), M(9, -2)