MAC 1114 Module 3 Radian Measure and Circular

MAC 1114 Module 3 Radian Measure and Circular Functions Rev. S 08

Learning Objectives Upon completing this module, you should be able to: 1. 2. 3. 4. 5. 6. 7. 8. Convert between degrees and radians. Find function values for angles in radians. Find arc length on a circle. Find area of a sector of a circle. Solve applications. Define circular functions. Find exact circular function values. Approximate circular function values. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 2

Radian Measure and Circular Functions There are three major topics in this module: - Radian Measure - Applications of Radian Measure - The Unit Circle and Circular Functions Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 3

Introduction to Radian Measure n 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. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 4

How to Convert Between Degrees and Radians? n 1. Multiply a degree measure by simplify to convert to radians. n 2. Multiply a radian measure by to convert to degrees. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. radian and simplify 5

Example of Converting from Degrees to Radians n Convert each degree measure to radians. a) 60° n b) 221. 7° n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 6

Example of Converting from Radians to Degrees n Convert each radian measure to degrees. n a) n b) 3. 25 Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 7

Let’s Look at Some Equivalent Angles in Degrees and Radians Degrees Radians Exact 0° Approximate Radians Exact 0 90° 30° . 52 180° 45° . 79 270° 60° 1. 05 360° Rev. S 08 0 Degrees http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. Approximate 1. 57 3. 14 π 4. 71 6. 28 2π 8

Let’s Look at Some Equivalent Angles in Degrees and Radians (cont. ) Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 9

Examples n n n Find each function value. a) n b) Convert radians to degrees. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 10

How to Find Arc Length of a Circle? n The length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle, or s = rθ , θ in radians. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 11

Example of Finding Arc Length of a Circle n 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) n b) 144° n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 12

Example of Finding Arc Length of a Circle (cont. ) n a) r = 18. 2 cm and θ = n b) convert 144° to radians n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 13

Example of Application n 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°? n Rev. S 08 Convert 39. 72 to radian measure. http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 14

Let’s Practice Another Application of Radian Measure Problem n 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? Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 15

Let’s Practice Another Application of Radian Measure Problem (cont. ) n 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. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 16

Let’s Practice Another Application of Radian Measure Problem (cont. ) n An arc with this length on the larger gear corresponds to an angle measure θ , in radians where n Convert back to degrees. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 17

How to Find Area of a Sector of a Circle? n A sector of a circle is a portion of the interior of a circle intercepted by a central angle. “A piece of pie. ” n The area of a sector of a circle of radius r and central angle θ is given by Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 18

Example n n n Find the area of a sector with radius 12. 7 cm and angle θ = 74°. Convert 74° to radians. Use the formula to find the area of the sector of a circle. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 19

What is a Unit Circle? n A unit circle has its center at the origin and a radius of 1 unit. Note: r = 1 ns = rθ , ns=θ in radians. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 20

Circular Functions Note that s is the arc length measured in linear units such as inches or centimeters, is numerically equal to the angle θ measured in radians, because r = 1 in the unit circle. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 21

Let’s Look at the Unit Circle Again Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 22

What are the Domains of the Circular Functions? n Assume that n is any integer and s is a real number. Sine and Cosine Functions: (−∞, ∞) n Tangent and Secant Functions: n Cotangent and Cosecant Functions: n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 23

How to Evaluate a Circular Function? n 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. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 24

Example of Finding Exact Circular Function Values n n n 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 n Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 25

Example of Approximating Circular Function Values n n q n Find a calculator approximation to four decimal places for each circular function. (Make sure the calculator is in radian mode. ) a) cos 2. 01 ≈ −. 4252 b) cos. 6207 ≈. 8135 For the cotangent, secant, and cosecant functions values, we must use the appropriate reciprocal functions. c) cot 1. 2071 Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 26

What have we learned? We have learned to: 1. 2. 3. 4. 5. 6. 7. 8. Convert between degrees and radians. Find function values for angles in radians. Find arc length on a circle. Find area of a sector of a circle. Solve applications. Define circular functions. Find exact circular function values. Approximate circular function values. Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 27

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, 8 th Edition Rev. S 08 http: //faculty. valenciacc. edu/ashaw/ Click link to download other modules. 28