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5.4 Radians, Arc Length, Angular Speed Mon Jan 06

5.4 Radians, Arc Length, Angular Speed Mon Jan 06. 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

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5.4 Radians, Arc Length, Angular Speed Mon Jan 06

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  1. 5.4 Radians, Arc Length, Angular SpeedMon Jan 06 Do Now Find the following: 1) Cos 150 deg 2) Tan 210 deg

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

  3. 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

  4. Unit Circle and Arc Length • In the unit circle, the radius is 1 • Therefore, the circumference of a unit circle is 2pi

  5. 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?

  6. 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)

  7. 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

  8. 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

  9. Converting between degrees and radians • To convert from degree to radians, • To convert from radians to degrees,

  10. Ex • Convert to radians • 1) 120 deg • 2) -297.25 deg

  11. Ex • Convert to degrees • 1) • 2) 8.5

  12. 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!

  13. Unit circle and radians • Draw degrees and radians in the unit circle here

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

  15. Ex • Find the complement and supplement of pi/6

  16. Closure • Convert 315 deg to radians • Convert pi/3 to degrees • HW: p.486 #1-59 odds

  17. 5.4 Arc Length and Central Angles, Angular SpeedTues Jan 07 • Do Now • Find the following values • 1) sin pi/4 • 2) tan 7pi/6

  18. HW Review: p.486 #1-59

  19. 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:

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

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

  22. 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

  23. 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

  24. 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

  25. 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

  26. Ex2 • An anchor is hoisted at a rate of 2ft/sec as the chain is woun around a capstan with a 1.8yd diameter. What is the angular speed of the capstan?

  27. 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

  28. 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 Thurs

  29. 5.3-5.4 ReviewWed Jan 09 • 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?

  30. HW Review: p.488 #61-79

  31. 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 • Trig values in radians • Linear and Angular speed

  32. Closure • What is a radian? How are radians and degrees related? • 5.3-5.4 Quiz Thurs (maybe Fri?)

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