1. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-1 Radian Measure 3.1 Radian Measure ▪ Converting Between Degrees and Radians ▪ Finding Function Values for Angles in Radians

2. 3.1 Conversion for Degrees ↔ Radians Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-2 Given a circle with a radius of 1 What is the circumference? Circumference = 2πr = 2π The degrees all the way around a circle is 360˚ 2π radians = 360˚ π radians = 180˚

3. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-3 Convert each degree measure to radians. 3.1 Example 1 Converting Degrees to Radians (page 103) Convert each radian measure to degrees. ˚ ˚ ˚ ˚

4. 3-4 Find each function value. 3.1 Example 3 Finding Function Values of Angles in Radian Measure (page 105) Ref angle is 30˚ Ref angle is 60˚ Ref angle is 45˚ 1 2 1 2 1

5. 3.1 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-5 Radian Measure Converting Between Degrees and Radians Finding Function Values for Angles in Radians π radians = 180˚

6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-6 Applications of Radian Measure 3.2 Arc Length on a Circle ▪ Area of a Sector of a Circle

7. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-7 A circle has radius 25.60 cm. Find the length of the arc intercepted by a central angle having each of the following measures. 3.2 Example 1 Finding Arc Length Using s = rθ (page 109) Convert θ to radians. Could have used: fraction of circumference ( n˚ / 360˚ )2πr

8. 3-8 Erie, Pennsylvania is approximately due north of Columbia, South Carolina. The latitude of Erie is 42° N, while that of Columbia is 34° N. Find the north-south distance between the two cities. 3.2 Ex 2 Using Latitudes to Find the Distance Between Two Cities (page 110) The radius of the earth is about 6400 km. The central angle between Erie and Columbia is 42° – 34° = 8°. Convert 8° to radians: Use s = rθ to find the north-south distance between the two cities. The north-south distance between Erie and Columbia is about 893.61km 893.61km IWHD: s = ( 8˚ / 360˚ )2π6400 Side note: What if one city is 42˚N and the other 34˚S?

9. 3-9 A rope is being wound around a drum with radius 0.327 m. How much rope will be wound around the drum if the drum is rotated through an angle of 132.6°? 3.2 Example 3 Finding a Length Using s = rθ (page 110) The length of rope wound around the drum is the arc length for a circle of radius 0.327 m and a central angle of 132.6°. Convert 132.6° to radians: Use s = rθ to find the arc length, which is the length of the rope. The length of the rope wound around the drum is about 0.757 m. IWHD: s = ( 132.6˚ / 360˚ )2π0.327

10. Two gears are adjusted so that the smaller gear drives the larger one. If the radii of the gears are 3.6 in. and 5.4 in., and the smaller gear rotates through 150°, through how many degrees will the larger gear rotate? 3.2 Example 4 Finding an Angle Measure Using s = rθ (pg 111) The arc length traveled for the smaller gear with a central angle of 150˚ will be equal to the arc length of the larger gear through what central angle? The larger gear will rotate through 100°. 3-10

11. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-11 Find the area of a sector of a circle having radius 15.20 ft and central angle 108.0°. 3.2 Example 5 Finding the Area of a Sector (page 112) The area of the sector is about 217.8 sq ft. Could have used: Fraction of the area ( n˚ / 360˚ )πr 2 IWHD: A = ( 108.0˚ / 360˚ )π15.2 2

12. 3.2 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-12 Arc Length s = rθ or s = ( n˚ / 360˚ )2πr Area of a Sector A = ½ r 2 θ or A = ( n˚ / 360˚ )πr 2

13. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-13 The Unit Circle and Circular Functions 3.3 Circular Functions ▪ Finding Values of Circular Functions ▪ Determining a Number with a Given Circular Function Value

14. 3.3 Radian Measures’ on the Axis Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-14 0 π/2π/2 π 3π/23π/2 2π2π 5π/25π/2 3π3π 7π/27π/2 4π4π 9π/29π/2 5π5π And the pattern continues…

15. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-15 Find the exact values of sin (–3 π ), cos (–3 π ), and tan (–3 π ). 3.3 Example 1 Finding Exact Circular Function Values (page 122) An angle of –3 π intersects the unit circle at (–1, 0). Reminder: Sin θ = y / r Cos θ = x / r Tan θ = y / x

16. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-16 Use reference angles and degree/radian conversion to find the exact value of 3.3 Example 2(c) Finding Exact Circular Function Values (page 122) In standard position, 330° lies in quadrant IV with a reference angle of 30°, so 30˚ 1 2 √3 A C T S

17. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-17 Use the figure to find the exact values of 3.3 Example 2(a) Finding Exact Circular Function Values (page 122) Sin θ = y / r Cos θ = x / r

18. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-18 3.3 Example 2(b) Finding Exact Circular Function Values (page 122) Use the figure and the definition of tangent to find the exact value of

19. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-19 Find a calculator approximation for each circular function value. 3.3 Example 3 Approximating Circular Function Values (page 123) (a)sin 3.42 (b)tan.8234 ≈ –.2748 ≈ 1.0790 (c)sec 5.6041≈ 1.2851 (d)csc(–2.7335)≈ –2.5198 sec 5.6041 = 1/cos 5.6041 = csc (-2.7335) = 1/sin(-2.7335)

20. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-20 Approximate the value of s in the interval if sin s =.3210. 3.3 Example 4(a) Finding a Number Given Its Circular Function Value (page 123) Use the inverse sine function of a calculator. Anytime you are looking for the angle measure, remember to use inverse sin or inverse cos or inverse tan… And ask yourself, is the answer supposed to be in radians or degrees and what mode is your calculator in? Answer: s = 0.3268 radians

21. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-21 Find the value of s in the interval if 3.3 Example 4(b) Finding a Number Given Its Circular Function Value (page 123) They want the angle measure between 3π/2 and 2π. This is in quadrant IV and they want the final answer in radians. This one will be on NO calculator part. 30˚ 1 2 √3 60˚ A reference angle of 30˚ in quadrant IV is equivalent to an angle of 330˚ Then convert 330˚ to radians and get s = 11π / 6

22. 3.3 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-22 The Unit Circle and Circular Functions Circular Functions Finding Values of Circular Functions Determining a Number with a Given Circular Function Value

23. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-23 Linear and Angular Speed 3.4 Linear Speed ▪ Angular Speed

24. 3.4 Formulas to Memorize for this section Linear Formulas: D = R T S = V T V = S / T 3-24 Circular Formulas: ω is the angular speed that an object is traveling around a circular path measured in radians per unit of time θ is the number of radians traveled by an object around a circular path θ = ωt = ( radians / time )(time) = radians S = rθ S is the distance traveled around the circular path (arc length) One more: v = s / t v = rθ / t v = r(ωt) / t v = rω V = rω Therefore, the linear velocity of an object traveling in a circular path is

25. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-25 Suppose that P is on a circle with radius 15 in., and ray OP is rotating with angular speed radian per second. 3.4 Ex 1 Using Linear and Angular Speed Formulas (page 130) (a)Find the angle generated by P in 10 seconds. D = RT S = VT Linear distance = linear velocity x time S = rθ Circular distance = radius x radians (arc length) θ is the # of radians traveled in a circular path ω is the angular velocity (radians per unit of time) θ = ωt = ( rad / time ) time V = rω Linear velocity = radius x angular velocity find θ t = 10 ω = π / 12 r = 15 = π / 12 10 = 5π / 6 radians θ = ωt

26. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-26 3.4 Ex 1 Finding Exact Circular Function Values (cont.) (b)Find the distance traveled by P along the circle in 10 seconds. Suppose that P is on a circle with radius 15 in., and ray OP is rotating with angular speed radian per second. find s t = 10 ω = π / 12 r = 15 D = RT S = VT Linear distance = linear velocity x time S = rθ Circular distance = radius x radians (arc length) θ is the # of radians traveled in a circular path ω is the angular velocity (radians per unit of time) θ = ωt = ( rad / time ) time V = rω Linear velocity = radius x angular velocity =rωt =15 π / 12 10 = 25π / 2 in ≈ 39.9 in

27. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-27 3.4 Ex 1 Finding Exact Circular Function Values (cont.) (c)Find the linear speed of P in inches per second for a 10 second interval. Suppose that P is on a circle with radius 15 in., and ray OP is rotating with angular speed radian per second. find v t = 10 ω = π / 12 r = 15 D = RT S = VT Linear distance = linear velocity x time S = rθ Circular distance = radius x radians (arc length) θ is the # of radians traveled in a circular path ω is the angular velocity (radians per unit of time) θ = ωt = ( rad / time ) time V = rω Linear velocity = radius x angular velocity V = rω= 15 π / 12 = 5π / 4 in/sec

28. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-28 A belt runs a pulley of radius 5 inches at 120 revolutions per minute. 3.4 Ex 2 Finding Angular Speed of a Pulley and Linear Speed of a Belt (page 130) (a)Find the angular speed of the pulley in radians per second. D = RT S = VT Linear distance = linear velocity x time S = rθ Circular distance = radius x radians (arc length) θ is the # of radians traveled in a circular path ω is the angular velocity (radians per unit of time) θ = ωt = ( rad / time ) time V = rω Linear velocity = radius x angular velocity ω = 1202π/min = 240π rad/minr = 5

29. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 3-29 A belt runs a pulley of radius 5 inches at 120 revolutions per minute. (b)Find the linear speed of the belt in inches per second. 3.4 Ex 2 Finding Angular Speed of a Pulley and Linear Speed of a Belt (cont) find v r = 5 ω = 1202π/min = 240π rad/min = 4π rad/sec D = RT S = VT Linear distance = linear velocity x time S = rθ Circular distance = radius x radians (arc length) θ is the # of radians traveled in a circular path ω is the angular velocity (radians per unit of time) θ = ωt = ( rad / time ) time V = rω Linear velocity = radius x angular velocity

30. 3-30 A satellite traveling in a circular orbit approximately 1800 km above the surface of Earth takes 2.5 hours to make an orbit. 3.4 Ex 3 Finding Linear Speed and Distance Traveled by a Satellite (page 131) (a)Approximate the linear speed of the satellite in kilometers per hr. The distance of the satellite from the center of Earth: r ≈ 1800 + 6400 = 8200 km. v = rω v = 8200 km ( 4π / 5 ) rad / hr = 6560π km / hr ≈ 20608.85 km / hr find v r = 8200 km D = RT S = VT Linear distance = linear velocity x time S = rθ Circular distance = radius x radians (arc length) θ is the # of radians traveled in a circular path ω is the angular velocity (radians per unit of time) θ = ωt = ( rad / time ) time V = rω Linear velocity = radius x angular velocity t = 2.5 hrs ω = 2π/(2.5 hrs) = 4 / 5 π rad/hr

31. 3-31 A satellite traveling in a circular orbit approximately 1800 km above the surface of Earth takes 2.5 hours to make an orbit. 3.4 Ex 3 Finding Linear Speed and Distance Traveled by a Satellite cont (b)Approximate the distance the satellite travels in 3.5 hours. Linear Velocity from previous slide is 6560π km / hr

32. 3.4 Formulas to Memorize for this section Linear Formulas: D = R T S = V T V = S / T 3-32 Circular Formulas: ω is the angular speed that an object is traveling around a circular path measured in radians per unit of time θ is the number of radians traveled by an object around a circular path θ = ωt = ( radians / time )(time) = radians S = rθ S is the distance traveled around the circular path V = rω The linear velocity of an object traveling in a circular path is