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Aim: What are radians and how do they differ from degrees?

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Aim: What are radians and how do they differ from degrees?

Do Now:

A kite is held by a taut string pegged to the ground. If the string is 40 feet long and makes a 330 angle with the ground, how high is the kite?

h

h

Angle of elevation

40’

40’

330

A kite is held by a taut string pegged to the ground. If the string is 40 feet long and makes a 330 angle with the ground, how high is the kite?

Sin 330 =

.544639. . . =

40 feet

h 21.8 feet

The kite is approximately 22 feet off the ground.

Circle - Central Angles

B

900

A

O

1800

Central Angle – of a circle is an angle

whose vertex is the center of the circle.

The sum of all central angles of a circle is 3600.

Arc AB = r

1 radian

If the length of arc AB

measures r, then the

measure of central angle

BOA is 1 radian.

1

2

3

4

5

6

.28

RadiansDefinition: A radian is the measure of the

central angle that intercepts an arc equal

in length to the radius of the circle.

B

r

A

r

O

- radians = 1800 2 radians = 3600

r

2 units

2

2

1

2

1

2

2

Radians - Unit CircleIf the measure of arc AB

is 1 unit, then the

measure of central angle

BOA is 1 radian.

Given: radius = 1

Arc AB = 1

B

1

1 radian

A

1

O

length of radius

Finding the Measure of Central Angles in RadiansB

s

A

r

O

measure of angle in radians =

- Arc AB defines
- a semi-circle
- with
- length s.
- is the central

angle whose measure

is 1800.

B

A

r

O

s = 1/2 C =

Radians & Degrees - SemicircleC = pD = p2r

1/2 (2πr) = πr

= 1800

length of intercepted arc s

radians = 2700

length of radius r

Measure of Central Angles in Radiansmeasure of angle in radians =

B

s

A

O

r

π radians = 1800

2π radians = 3600

What is the complement of = /12?

radians = 900

Supplementary, Complementary & Coterminal Anglesπ radians = 1800

2π radians = 3600

What is the supplement of = 5/6?

Name, in terms of radians, a coterminal angle for = 17/6

mCentral = 1.5 radians

substitute the

given values

Model ProblemIn a circle, the length of a radius is 4

centimeters. Find the length of an arc

intercepted by a central angle whose

measure is 1.5 radians.

solve for s

s = 6 cm.

Model Problem

A weather satellite in a circular orbit around Earth completes one orbit every 3 hours. The radius of the Earth is about 6400 km, and the satellite is positioned 2600 km above Earth. How far does the satellite travel in 1 hour.

s

since one complete

rotation takes 3 hr.,

the satellite completes

1/3 of a rotation in 1hr.

r

s = distance traveled in 1 hr.

r = 6400 + 2600 = 9000

x = degrees

solve for x:

Changing from Radians to DegreesHow do we convert a π/4 radians to degrees?

Method 1

πx = 45π

x = 45 degrees

measure of a straight angle in degrees

=

mA in radians

measure of a straight angle in radians

Proportion:

identify variable:

x = measure in

radians of 750 &

set up proportion

solve for x

Degrees to Radians – Model ProblemConvert a 750 to radians?

Use the fact: π radians = 1800

180x = 75π

Method 1

Proportion:

x = degrees in

angle of 7π/3 radians

solve for x:

Model Problemπx = 420π

x = 420 degrees

Proportion:

identify variable:

x = measure in

radians of 1350 &

set up proportion

solve for x

Model ProblemsConvert 1350 to radians?

180x = 135π

Regents Prep

What is the number of degrees in an angle whose radian measure is

- 150 2. 165
- 3. 330 4. 518

Find to the nearest minute, the angle whose measure is 3.45 radians.

x = 197.6704o

x = 197o 40’

convert .6704o to minutes

m’ = 40.2’

Model Problem

In a circle, a central angle of 1/3 radian intercepts an arc of 3 centimeters. Find the length, in centimeters, of a radius of the circle.

If is 4 and r = 1.25, find s.

Model Problem

If f(x) = cos 2x and sin x, find f(/2)

substitute

simplify

evaluate

If f(x) = sinx cos2x, find f(/3)

Finding Arc Length

A circle has a radius of 4 inches. Find the length of an arc cut off by a central angle of 2400.

substitute r = 4 into s = r

Regents Prep

A circle has a radius of 4 inches. In inches, what is the length of the arc intercepted by a central angle of 2 radians?

1. 2 2. 2 3. 8 4. 8

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