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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 9 p • 1) 4 • 2) 7p 6 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, deg × p rad 180° = radians • To convert from radians to degrees, 180° radians × = degrees p rad Ex • Convert to radians • 1) 120 deg • 2) -297.25 deg Ex • Convert to degrees • 1) 3p 4 • 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: S q= 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 s v= t • Angular Speed is defined as amount of rotation per unit of time q w= t 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 v = rw • Where w 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 – 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