3.2 applications of radian...

Copyright © 2009 Pearson Addison-Wesley 1.1-1 3.2-1

Applications of Radian

Measure 3.2

• Find the arc length on a circle. • Find the area of a sector of a circle. • Use latitudes to find the distance between two locations.


The length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or

Arc Length

s = rθ, θ in radians


Caution

Avoid the common error of applying

this formula with θ in degrees.

When applying the formula s = rθ,

the value of θ MUST be in radians.


Example 1(a) FINDING A LENGTH USING s = rθ

A rope is being wound around a drum with radius

.8725 ft. How much rope will be wound around the

drum if the drum is rotated through an angle of

39.72°?

The length of rope wound around the drum is the arc

length for a circle of radius .8725 ft and a central angle

of 39.72°.


Use s = rθ to find the arc length, which is the length of

the rope. Remember to convert 39.72° to radians

The length of the rope wound around the drum is

about .6049 ft.

Example 1(a) FINDING A LENGTH USING s = rθ

(continued)


Find the length of the radius of a circle if the arc

intercepted by a central angle with measure 144°

has a length of 45.7 cm.

Example 1(b) FINDING ARC LENGTH USING s = rθ

Convert θ to radians.

Approximate Lengths of Chords

If we are working with relatively small central angles in circles with large radii, then we can use the arc length to approximate the length of the associated chord.

Length chord AB ≈ Length arc AB

Copyright © 2009 Pearson Addison-Wesley 3.2-7


A person standing on the earth notices that a 747

Jumbo Jet flying overhead subtends an angle of

0.45°. If the length of the jet is 230 ft, find its

altitude to the nearest thousand feet.

𝑠 = 230

𝜃 = .45𝜋

180=

𝜋

400

230 = 𝑟𝜋

400

𝒓 ≈ 𝟐𝟗, 𝟎𝟎𝟎 ft

Example 2 APPROXIMATING THE LENGTH OF A

CHORD USING ARC LENGTH

230 ft

.45°

r

Latitude

Latitude gives the measure of a central angle with

vertex at the Earth’s center whose initial side is

the equator and whose terminal side goes through

the given location.

Right now we will only be able to find the distance

between two locations on the same north-south

line.



Example 3 USING LATITUDES TO FIND THE

DISTANCE BETWEEN TWO CITIES

Reno, Nevada is approximately due north of Los

Angeles. The latitude of Reno is 40° N, while that of

Los Angeles is 34° N. The radius of Earth is about

6400 km. Find the north-south distance between the

two cities.

The central angle between

Reno and Los Angeles is

40° – 34° = 6°. Convert 6° to

radians:




(continued)

Use s = rθ to find the north-south distance between

the two cities.

The north-south distance

between Reno and Los Angeles

is about 670 km.




Assume the radius of Earth is about 6400 km. Spokane, WA and Jordan Valley, OR are 486 km apart and lie on the same north-south line. The latitude of Spokane is 43º N. What is the latitude of Jordan Valley?

First find the angle between Spokane and Jordan Valley.

Convert the angle to degrees.

Subtract the angle from the latitude of Spokane.


Example 5 FINDING AN ANGLE MEASURE USING

s = rθ

Two gears are adjusted so that the smaller gear drives

the larger one. If the smaller gear rotates through an

angle of 225°, through how many degrees will the

larger gear rotate?

First find the radian measure

of the angle, and then find the

arc length on the smaller gear

that determines the motion of

the larger gear.



s = rθ (continued)

An arc with this length on the

larger gear corresponds to an

angle measure θ:

The arc length on the smaller gear is



s = rθ (continued)

The larger gear rotates

through an angle of 117°.

Convert θ to degrees:

Copyright © 2009 Pearson Addison-Wesley 3.2-16

Area of a Sector of a Circle

A sector of a circle is a portion of the interior of a

circle intercepted by a central angle.

Think of it as a “ piece of pie.”


Area of a Sector

The area A of a sector of a circle of radius r and central angle θ is given by

Caution

The value of θ must be in radians

when using the formula for the area

of a sector.


Example 6 FINDING THE AREA OF A SECTOR

A lawn sprinkler located at the corner of a yard is set

to rotate through 90° and project water out 30 ft.

What area of the lawn is watered by the sprinkler?

Give your answer in terms of 𝝅.

𝑟 = 30 𝐴 =1

2(30)2 𝜋

2

𝜃 = 90° =𝜋

2 𝑨 = 𝟐𝟐𝟓𝝅 ≈ 𝟕𝟎𝟔. 𝟗 𝐟𝐭𝟐

Find the area of the sector.



If the sector formed by a central angle of 15° has

an area of 𝜋

3 cm2 , find the radius of the circle.

Round your answer to the nearest tenth.

𝜃 = 15°𝜋

180=

𝜋

12

𝜋

3=

1

2𝑟2 𝜋

12

𝐴 =𝜋

3 8 =

1

2𝑟2

𝑟 =? 𝑟2 = 16

𝒓 = 𝟒



Assignment

p. 113 (2 – 12 even, 18 – 34 even, 40 – 48 even,

54 – 58 even)


3.2 applications of radian...

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