1. Rev.S08 MAC 1114 Module 3 Radian Measure and Circular Functions

2. 2 Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. Convert between degrees and radians. 2. Find function values for angles in radians. 3. Find arc length on a circle. 4. Find area of a sector of a circle. 5. Solve applications. 6. Define circular functions. 7. Find exact circular function values. 8. Approximate circular function values. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

3. 3 Rev.S08 Radian Measure and Circular Functions http://faculty.valenciacc.edu/ashaw/ http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. - Radian Measure - Applications of Radian Measure - The Unit Circle and Circular Functions There are three major topics in this module:

4. 4 Rev.S08 Introduction to Radian Measure http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. 5 Rev.S08 How to Convert Between Degrees and Radians? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. 6 Rev.S08 Example of Converting from Degrees to Radians http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Convert each degree measure to radians. a)60  b) 221.7 

7. 7 Rev.S08 Example of Converting from Radians to Degrees http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Convert each radian measure to degrees. a) b) 3.25

8. 8 Rev.S08 Let’s Look at Some Equivalent Angles in Degrees and Radians http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. 9 Rev.S08 Let’s Look at Some Equivalent Angles in Degrees and Radians (cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

10. 10 Rev.S08 Examples http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Find each function value. a) Convert radians to degrees. b)

11. 11 Rev.S08 How to Find Arc Length of a Circle? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

12. 12 Rev.S08 Example of Finding Arc Length of a Circle http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 

13. 13 Rev.S08 Example of Finding Arc Length of a Circle (cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. a) r = 18.2 cm and  = b) convert 144  to radians

14. 14 Rev.S08 Example of Application http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

15. 15 Rev.S08 Let’s Practice Another Application of Radian Measure Problem http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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?

16. 16 Rev.S08 Let’s Practice Another Application of Radian Measure Problem (cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

17. 17 Rev.S08 Let’s Practice Another Application of Radian Measure Problem (cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. An arc with this length on the larger gear corresponds to an angle measure , in radians where Convert back to degrees.

18. 18 Rev.S08 How to Find Area of a Sector of a Circle? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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

19. 19 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

20. 20 Rev.S08 What is a Unit Circle? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. A unit circle has its center at the origin and a radius of 1 unit. Note: r = 1 s = r , s=  in radians.

21. 21 Rev.S08 Circular Functions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Note that s is the arc length measured in linear units such as inches or centimeters, is numerically equal to the angle  measured in radians, because r = 1 in the unit circle.

22. 22 Rev.S08 Let’s Look at the Unit Circle Again http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

23. 23 Rev.S08 What are the Domains of the Circular Functions? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Assume that n is any integer and s is a real number. Sine and Cosine Functions: (−∞, ∞) Tangent and Secant Functions: Cotangent and Cosecant Functions:

24. 24 Rev.S08 How to Evaluate a Circular Function? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

25. 25 Rev.S08 Example of Finding Exact Circular Function Values http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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

26. 26 Rev.S08 Example of Approximating Circular Function Values http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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

27. 27 Rev.S08 What have we learned? We have learned to: 1. Convert between degrees and radians. 2. Find function values for angles in radians. 3. Find arc length on a circle. 4. Find area of a sector of a circle. 5. Solve applications. 6. Define circular functions. 7. Find exact circular function values. 8. Approximate circular function values. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

28. 28 Rev.S08 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.