1. Copyright © 2005 Pearson Education, Inc.

2. Chapter 3 Radian Measure and Circular Functions

3. Copyright © 2005 Pearson Education, Inc. 3.1 Radian Measure

4. Copyright © 2005 Pearson Education, Inc. Slide 3-4 Radian Measure An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian.

5. Copyright © 2005 Pearson Education, Inc. Slide 3-5 Converting Between Degrees and Radians 1. Multiply a degree measure by radian and simplify to convert to radians. 2.Multiply a radian measure by and simplify to convert to degrees.

6. Copyright © 2005 Pearson Education, Inc. Slide 3-6 Example: Degrees to Radians Convert each degree measure to radians. a)60  b) 221.7 

7. Copyright © 2005 Pearson Education, Inc. Slide 3-7 Example: Radians to Degrees Convert each radian measure to degrees. a) b) 3.25

8. Copyright © 2005 Pearson Education, Inc. Slide 3-8 Equivalent Angles in Degrees and Radians 6.28 22 360  1.05 60  4.71 270 .79 45  3.14  180 .52 30  1.57 90  00 00 ApproximateExactApproximateExact RadiansDegreesRadiansDegrees

9. Copyright © 2005 Pearson Education, Inc. Slide 3-9 Equivalent Angles in Degrees and Radians continued

10. Copyright © 2005 Pearson Education, Inc. Slide 3-10 Example: Finding Function Values of Angles in Radian Measure Find each function value. a) Convert radians to degrees. b)

11. Copyright © 2005 Pearson Education, Inc. 3.2 Applications of Radian Measure

12. Copyright © 2005 Pearson Education, Inc. Slide 3-12 Arc Length The length s of the arc intercepted on a circle of radius r by a central angle of measure  radians is given by the product of the radius and the radian measure of the angle, or s = r ,  in radians.

13. Copyright © 2005 Pearson Education, Inc. Slide 3-13 Example: Finding Arc Length A circle has radius 18.2 cm. Find the length of the arc intercepted by a central angle having each of the following measures. a) b) 144 

14. Copyright © 2005 Pearson Education, Inc. Slide 3-14 Example: Finding Arc Length continued a) r = 18.2 cm and  = b) convert 144  to radians

15. Copyright © 2005 Pearson Education, Inc. Slide 3-15 Example: Finding a Length A rope is being wound around a drum with radius.8725 ft. How much rope will be wound around the drum it the drum is rotated through an angle of 39.72  ? Convert 39.72 to radian measure.

16. Copyright © 2005 Pearson Education, Inc. Slide 3-16 Example: Finding an Angle Measure Two gears are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates through 225 , through how many degrees will the larger gear rotate?

17. Copyright © 2005 Pearson Education, Inc. Slide 3-17 Solution Find the radian measure of the angle and then find the arc length on the smaller gear that determines the motion of the larger gear.

18. Copyright © 2005 Pearson Education, Inc. Slide 3-18 Solution continued An arc with this length on the larger gear corresponds to an angle measure , in radians where Convert back to degrees.

19. Copyright © 2005 Pearson Education, Inc. Slide 3-19 Area of a Sector A sector of a circle is a portion of the interior of a circle intercepted by a central angle. “A piece of pie.” The area of a sector of a circle of radius r and central angle  is given by

20. Copyright © 2005 Pearson Education, Inc. Slide 3-20 Example: Area Find the area of a sector with radius 12.7 cm and angle  = 74 . Convert 74  to radians. Use the formula to find the area of the sector of a circle.

21. Copyright © 2005 Pearson Education, Inc. 3.3 The Unit Circle and Circular Functions

22. Copyright © 2005 Pearson Education, Inc. Slide 3-22 Unit Circle A unit circle has its center at the origin and a radius of 1 unit.

23. Copyright © 2005 Pearson Education, Inc. Slide 3-23 Circular Functions

24. Copyright © 2005 Pearson Education, Inc. Slide 3-24 Unit Circle--more

25. Copyright © 2005 Pearson Education, Inc. Slide 3-25 Domains of the Circular Functions Assume that n is any integer and s is a real number. Sine and Cosine Functions: ( ,  ) Tangent and Secant Functions: Cotangent and Cosecant Functions:

26. Copyright © 2005 Pearson Education, Inc. Slide 3-26 Evaluating a Circular Function Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations. Calculators must be in radian mode when finding circular function values.

27. Copyright © 2005 Pearson Education, Inc. Slide 3-27 Example: Finding Exact Circular Function Values Find the exact values of Evaluating a circular function at the real number is equivalent to evaluating it at radians. An angle of intersects the unit circle at the point. Since sin s = y, cos s = x, and

28. Copyright © 2005 Pearson Education, Inc. Slide 3-28 Example: Approximating Find a calculator approximation to four decimal places for each circular function. (Make sure the calculator is in radian mode.) a) cos 2.01  .4252b) cos.6207 .8135  For the cotangent, secant, and cosecant functions values, we must use the appropriate reciprocal functions. c) cot 1.2071

29. Copyright © 2005 Pearson Education, Inc. 3.4 Linear and Angular Speed

30. Copyright © 2005 Pearson Education, Inc. Slide 3-30 Angular and Linear Speed Angular Speed: the amount of rotation per unit of time, where  is the angle of rotation and t is the time. Linear Speed: distance traveled per unit of time

31. Copyright © 2005 Pearson Education, Inc. Slide 3-31 Formulas for Angular and Linear Speed ( in radians per unit time,  in radians) Linear SpeedAngular Speed

32. Copyright © 2005 Pearson Education, Inc. Slide 3-32 Example: Using the Formulas Suppose that point P is on a circle with radius 20 cm, and ray OP is rotating with angular speed radian per second. a) Find the angle generated by P in 6 sec. b) Find the distance traveled by P along the circle in 6 sec. c) Find the linear speed of P.

33. Copyright © 2005 Pearson Education, Inc. Slide 3-33 Solution: Find the angle. The speed of ray OP is radian per second.

34. Copyright © 2005 Pearson Education, Inc. Slide 3-34 Solution: Find the angle continued The distance traveled by P along the circle is

35. Copyright © 2005 Pearson Education, Inc. Slide 3-35 Solution: Find the angle continued linear speed

36. Copyright © 2005 Pearson Education, Inc. Slide 3-36 Example: A belt runs a pulley of radius 6 cm at 80 revolutions per min. a) Find the angular speed of the pulley in radians per second. 80(2  ) = 160  radians per minute. 60 sec = 1 min b) Find the linear speed of the belt in centimeters per second. The linear speed of the belt will be the same as that of a point on the circumference of the pulley.