5.4 Radians, Arc Length, Angular Speed Thurs Oct 30 Do Now Find 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 degrees OR 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 Speed Fri 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 Review Nov 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 5.4 Radians 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