radian measure radian degree - manhattan high · pdf file... a circle has a radius 15 cm. find...

Trigonometry Unit 3 Packet

Name____________________ Hr___

Radian Measure

EXAMPLE 1: Convert from radians to degrees.

a) 3

4

b)

5

2

c)

4

3

d)

11

6

EXAMPLE 2: Convert from degrees to radians.

a) 120°

b) 270°

c) 315°

d) 210°

EXAMPLE 3: Find the value of the functions:

a) 2

tan3

b)

3sin

2

c)

5cos

6

Radian Degree

6

4

3

2

1 radian = _________°


Section 3.2: Applications of Radian Measure

Converting degrees to radians:

EXAMPLE 1: Convert to radians.

a) 147° b) 37.9° c) 221°

Arc length:

EXAMPLE 1: A circle has a radius 15 cm. Find the length of the arc intercepted by a

central angle having the following measure:

a) 2

5

radians b)

EXAMPLE 2: Two gears are adjusted to that the smaller gear drives the larger one. If

the smaller gear rotates through 225°, through how many degrees will the larger gear

rotate?

Area of a Sector:

EXAMPLE 3: Find the area of the given sector.

Caution‼!

θ must be in

RADIANS


Linear and Angular Speed

Suppose that an object moves along a circle or radius at a constant speed. If is the

distance traveled along the circle in time , then:

As the object moves along the circle, let θ be the angle made in time . Then:

There is an interesting relationship between linear and angular speed: _______________

EXAMPLE 1: Suppose that point P is on a circle with radius 10 cm, and ray OP is rotating

with angular speed 18

radian per second.

a) Find the angle generated

by P in 6 sec.

b) Find the distance traveled

by P along the circle in 6

sec.

c) Find the linear speed of P.

EXAMPLE 2: A belt runs a pulley of radius 6 cm at 80 revolutions per minute.

a) Find the angular speed of the pulley

in radians per second.

b) Find the linear speed of the belt in

centimeters per second.

linear speed =

angular speed =


EXAMPLE 3: The shoulder joint can rotate at about 25 radians per second. If a golfer’s

arm is straight and the distance from the shoulder to the club head is 5 ft, estimate the

linear speed of the club head from shoulder rotation.

EXAMPLE 4: Suppose you have rented a paddle boat at Tuttle Creek Lake. The current in

the lake causes the circular paddle wheel with radius 4 feet to rotate at a speed of 10

revolutions per minute. What is the speed of Tuttle Creek’s current in miles per hour?


PRACTICE:

Convert from radians to degrees:

1. 8

3

2.

7

4

3.

5

6

Convert from degrees to radians:

4. 300° 5. 390° 6. 225°

Find the exact value of the expression WITHOUT a calculator.

7. sin3

8. cos

6

9. tan

4

10. 2

cot3

11. csc

2

12.

5tan

6

13. 5

tan3

14.

15. 3

sec4

16. Through how many radians will the hour hand on a clock rotate in:

a) 24 hours? b) 4 hours?


17. Find the exact length of the arc

shown.

18. Find the radius of the circle.

Find the length of the arc intercepted by the central angle in a circle of radius .

19.

20.

21. If the radius of a circle is doubled, how is the length of the arc intercepted by a

fixed central angle changed?

22. Radian measure simplifies many formulas, including the one for arc length, .

Give the corresponding formula when is measured in degrees instead of radians.

23. Find the distance in kilometers between Wichita, KS, , and Fort Worth, TX,

, assuming they lie on the same north-south line. (The radius of the earth is

6400 km.)

3

12 in 3π

3π


Find the area of a sector having radius and central angle . Round to the nearest tenth.

24.

25.

26. Find the measure (in radians) of a central angle of a sector with area 16 in² inside a

circle with radius 3 inches

27. Find the radius of a circle in which a central angle of 6

radian determines a sector

of area 64 m².

28. Suppose that a point on a circle with radius and ray is rotating with angular

speed . Find each of the following:

i. the angle generated by in time .

ii. the distance traveled by along the circle in time .

iii. the linear speed of

a)

b)


29. Use the formula for angular velocity t

to find the value of the missing

variable.

a)

b)

c)

30. Use the formula for linear velocity ( ) to find the value of the missing

variable.

a)

b)

31. Find the angular velocity (in radians per second) of a line from the center to the

edge of a DVD revolving 300 times per minute.

32. Find the angular velocity of the minute hand of a clock.


33. Watch the video found at https://www.youtube.com/watch?v=WehHFJki9yQ.

Name five of the cool ways trigonometry helps us in the real world.

EXPANSION 1: Engineers use the term grade to represent

of a right angle and

express it as a percent. For example, an angle of would be referred to as a 1% grade.

a) By what number should you multiply a grade (disregarding the % symbol) to convert

it to radians?

b) In a rapid-transit rail system, the maximum grade allowed between two stations is

3.5%. Express this angle in degrees and radians.

EXPANSION 2: Find the linear velocity of the tip of an airplane propeller long,

rotating 500 times per minute.

EXPANSION 3: A circular pulley is rotating about its center. Through how many radians

would it turn in:

a) 8 rotations

b) 30 rotations

EXPANSION 4: The phase of the moon is modeled by ( )

( ) and gives the

fraction of the moon’s face that is illuminated by the sun. Evaluate each expression and

interpret the result.

a) ( ) b) (

) c) (

)

https://www.youtube.com/watch?v=WehHFJki9yQ


APPLICATIONS:

1. The mean distance of the Moon from Earth is miles. Assume that

the orbit of the Moon around Earth is circular and that 1 revolution takes

27.3 days.

a) Find the angular speed of the moon’s rotation in radians per hour.

b) Find the linear speed of a rock on the moon’s equator due to the moon’s rotation.

Express your answer in miles per hour.

2. You are bored in chemistry class and begin to stare at the clock.

Because Mrs. Sapp has sparked your interest in radian measure, you

begin to daydream, “I wonder how many radians the minute hand of

the clock has moved.”

a) Through how many radians does the minute hand of a clock

rotate from 12:45pm to 1:25pm?

b) The minute hand of the clock is 6 inches long. How far does the tip of the

minute hand move in 15 minutes? In 25 minutes?


3. You and your friends are riding your bike to Chipotle for lunch. The

diameter of each wheel of your bicycle is 26 inches.

a) If you are traveling at a speed of 20 miles per hour on your bicycle,

through how many revolutions per minute are the wheels turning?

b) In your desperation to beat your friends to Chipotle and be the first in line, you

increase your speed to 25 miles per hour. How does this change your angular

velocity? Explain.

c) If you get a larger bike with a diameter of 30 inches, what happens to your

angular velocity? Explain.


4. The free-throw line on an NCAA basketball court is 12 ft wide. In

international competition, it is only about 11.81 ft.

a) How much longer is the half circle above the free-throw line on the NCAA

court?

b) LeBron James can spin a basketball at 150 rpm. Find the linear velocity of the ball

in mph if the radius of the basketball is 6 inches.

5. You have part time job mowing lawns during the summer. You know that in

order for your lawn mower to properly cut the grass, the blade must strike

the grass at a speed of at least 5000 in/s.

a) If the innermost part of the cutting edge is 6 in from the center of the

blade, how many revolutions per minute must the blade turn?

b) Suppose you purchase a new mower with a blade diameter of 19 inches. If the

outermost tip of the blade hits a rock while turning at the rpms from part (a), how

fast in miles per hour could the rock be hurled from the mower? (This is why you

should always wear eye protection!)


6. You decide to ride the Ferris wheel with your friends at the Riley County

Fair. You wonder how fast the Ferris wheel is traveling. You observe

that it takes 20 seconds to make a complete revolution. Your seat is 25

feet from the axle of the wheel.

a) What is your angular velocity in radians per minute?

b) What is your linear velocity in miles per hour?

c) If the Ferris wheel had a radius of 20 feet instead of 25 feet, how would it

affect the linear speed?

7. You are mowing lawns for your summer job. You are getting tired of

your old mower because it requires a lot of strength to yank the start

cord. In order for the engine to start, the pulley must turn at 180

rpm. The pulley has a radius of 2.4 inches.

a) How many radians per second must the pulley turn in order to start?

b) How many feet per second must you pull the cord to start the mower?


Unit Wrap-Up

NO Calculator concepts:

Convert between degrees and radians:

225° -315°

Find the EXACT trig values:

(

)

Calculator concepts:

Find arc length using . Remember, θ should always be in RADIANS!

EXAMPLE: If NYC, New York is located at 41° N and Lima, Peru is located at 12° S, what

is the distance between the two cities? (The radius of the earth is 6400 km.)

Find the area of a sector using

. Remember, θ should always be in

RADIANS!

EXAMPLE: Suppose that a windshield wiper is 10 inches long and rotates back and forth

through an angle of 95°. What is the area of the region cleaned?

Find Linear and Angular Speed using r and t

.

EXAMPLE: Find the linear speed of a point on the edge of a flywheel of radius 7 cm if the

flywheel is rotating 90 times per second.

radian measure radian degree - manhattan high · pdf file... a circle has a radius 15 cm. find...

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