5.4 Radians, Arc Length, Angular SpeedThurs Oct 30

Do NowFind the following:

1) Cos 150 deg

2) Tan 210 deg

HW Review: p.472 #39-69 83-105

Unit Circle

• A circle centered at the origin with a radius of length 1 is called a unit circle

• This is useful because in a unit circle, any reference triangles have hypotenuses of 1

Unit Circle and Arc Length

• In the unit circle, the radius is 1

• Therefore, the circumference of a unit circle is 2pi

Ex

• How far will a point travel if it goes• A) ¼ of the way around the unit circle?

• B) 1/12 of the way?

• C) 3/8 of the way?

Ex2

• On the unit circle, mark the point determined by each of the following real numbers

• Hint: Consider how much more or less than 1pi or 2pi we want to find

• 1)

• 2)

Radian Measure

• The radian is the other measure of an angle.

• The word radian comes from the word radius

• Measuring 1 radius along the circumference of the circle determines an angle whose measure is 1 radian

Radians and 360 degrees

• Since 360 degrees denotes a full circumference, and 2pi is the circumference of the unit circle,

• 2pi radians = 360 degreesOR

• 1pi radians = 180 degrees

Converting between degrees and radians

• To convert from degree to radians,

• To convert from radians to degrees,

Ex

• Convert to radians• 1) 120 deg

• 2) -297.25 deg

Ex

• Convert to degrees• 1)

• 2) 8.5

Note

• When a rotation or angle is given in radians, the word radians is optional and most often omitted.

• If no unit is given, it is understood to be in radians

• No degree symbol = radians!

Unit circle and radians

• Draw degrees and radians in the unit circle here

Ex

• Find a positive angle and negative angle that are coterminal with 2pi/3

Ex

• Find the complement and supplement of pi/6

Closure

• Convert 315 deg to radians• Convert pi/3 to degrees

• HW: p.486 #1-59 odds

5.4 Arc Length and Central Angles, Angular SpeedFri Oct 31

• Do Now• Find the following values• 1) sin 3pi/4

• 2) cos 7pi/6

HW Review: p.486 #1-59

Radian Measure of other circles

• Radian measure can be determined using a circle other than a unit circle

• The radian measure of a rotation of the distance S traveled by a point at a radius R:

Ex

• Find the measure of a rotation in radians when a point 2m from the center of rotation travels 4m

Ex

• Find the length of an arc of a circle of radius 5cm associated with an angle of pi/3

Linear and Angular Speed

• Linear speed is defined as distance traveled per unit of time

• Angular Speed is defined as amount of rotation per unit of time

Linear Speed in terms of angular speed

• The linear speed v of a point a distance r from the center of rotation is given by

• Where is the angular speed in radians per unit of time

Note

• With these problems, all units must be the same (distance, radians, time)

• You may have to convert your given values in order to use the previous formula

Ex

• An earth satellite in circular orbit 1200km high makes one complete revolution every 90 min. What is its linear speed? Let the radius of the earth = 6400km

Ex2

• An anchor is hoisted at a rate of 2ft/sec as the chain is wound around a capstan with a 1.8yd diameter. What is the angular speed of the capstan?

Ex3

• A car is traveling at a speed of 70 mph. Its tires have an outside diameter of 28.56in. Find the angle through which a tire turns in 10 sec

Closure

• In a circle with a 5m radius, how long is an arc associated with an angle of 2.1 radians?

• HW: p.488 #61-79 odds• 5.3-5.4 quiz Tues

5.3-5.4 ReviewNov 03

• Do Now• A wheel with a 30cm radius is rotating at a

rate of 3radians/sec. What is the linear speed of a point on its rim, in meters per minute?

69

• Through how many radians does the minute hand of aclock rotate from 12:40 – 1:30

71

• A flywheel with a 15-cm diameter is rotating at a rate of 7radians/sec. What is the linear speed of a point on its rim, in cm/min

73

• A cylinder on this pres has a 21 in diameter. The linear speed of a point on the cylinder’s surface is 18.33 ft/sec. What is the angular speed of the cylinder in revolutions per hour?

HW Review: p.488 #61-79

5.3-5.4 Review

• 5.3 Trig Values of any angle– 4 Quadrants– Reference Angles – Unit Circle

• 5.4 Radians– Convert from/to degrees– Coterminal angles – Linear and Angular speed (1)

Do Now

• What is a radian? How are radians and degrees related?

• Length of an arc equivalent to the radius

• 2pi = 360 degrees• Pi = 180 degrees