Chapter 3. Radian Measure and the Unit Circle. For exercises 23 28, answers may vary

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1 Chapter Radian Measure and the Unit Circle Section Radian Measure For exercises 8, answers may vary.. Multiply the degree measure by radian 80 and reduce. Your answer will be in radians. Leave the answer as a multiple of, unless otherwise directed.. Multiply the radian measure by 80 and reduce. Your answer will be in degrees.. One radian is the measure of 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.. The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. A measure of one degree is equivalent to a rotation of /0 of a complete revolution. Thus, degree measure is based on the rotation of the terminal side of the angle, while radian measure is based on the length of the arc that is intercepted by the angle. (See exercise.) 7. A right angle measures 90 and intercepts an arc that is onequarter of the circumference of a circle, or =. Since the angle measures are equal, we have æ ö = x ç = x. çè ø 8. An angle of radian measure t in standard position intercepts an arc of length t on a circle of radius by definition. (See exercise.) Copyright 0 Pearson Education, Inc.

2 Section. Radian Measure Without the degree symbol on the 0, it is assumed that 0 is measured in radians. Thus, the approximate value of sin 0 is , not An angle of one radian is the measure of an angle in standard position that intercepts an arc that is equal to the length of the radius of the circle. In a unit circle, that length is. Copyright 0 Pearson Education, Inc.

3 Chapter Radian Measure and the Unit Circle 87. Begin the calculation with the blank next to 0º, and then proceed counterclockwise from there. ; ; ; 0 ; ; ; ; 7 ; 7 ; 0 ; 00 ; ; (a) 90. (a) 9. (a) 8 9. (a) (a) 00. ; 0.0 radian Section. Applications of Radian Measure To find the degree measure of a central angle in a circle if both the raidus and the length of the intercepted arc are known, first apply the formula s = rθ to find the radian measure. Then multiply the radian measure by 80 to find the degree measure...8 cm..08 cm.. ft..9 mi..0 m. 9 cm. 7.. in ft 9. The length of the arc is doubled. θ r km. 00 km. 900 km km. N.. N. 7. º 8. º cm.. 9. in.. (a). in cm.. in. (a) 9, rotations.9 mph; Yes 7. in 8. in 9. 7 in 0. 9 in. 0.0 km. 80 ft. Copyright 0 Pearson Education, Inc.

4 Section. The Unit Circle and Circular Functions º 8. 0º In Exercises 8, we will be rounding to the nearest tenth... m. 7.8 km ft. 0,0.9 yd..0 cm..m mi 8. 9,08. km m. 800 yd. the new area is twice the original area. 0 in.. (a) ft. (c) 7.8 ft (d) 7 ft 7. in. 7. (a) 0 ft 0 ft (c) ft m; 800 m yd 70.. mi 7. radius: 90 mi;circumference,800 mi 7. (a) The longitude at Greenwich is 0º. Answers will vary. 7. the area is quadrupled r θ sector =, θ is in degrees. 0 r θ h = or V = V r θ h 7. = θ ( ) 77. V r r h L r = θ θ 78. h= rcos æ θ ö 79. d = r ç cos çè ø L æ θ ö 80. d = ç cos θ çè ø Section.. (a) 0 (c) undefined. (a) 0 (c) 0. (a) 0 (c) 0. (a) 0 (c) 0. (a) 0 (c) 0. (a) 0 (c) undefined The Unit Circle and Circular Functions Copyright 0 Pearson Education, Inc.

5 Chapter Radian Measure and the Unit Circle sin θ = ; cos θ = ;tanθ = cotθ = ;sec θ = ; cscθ = cos cos 0.» sin» sin sin.8» cos.».0. θ». radians or θ» radians. θ». radians or θ».0 radians. θ» 0.8 radian or θ». radians. θ». radians or θ».0 radians. negative. negative 7. negative 8. positive 9. positive 0. negative... sin θ = ; cosθ = tanθ = ; cotθ = secθ = ;cscθ = 8 8 sin θ ; cos θ = ;tanθ = cot θ = ;sec θ = ; cscθ = 8 8 sin θ = ; cos θ = ; tanθ = cot θ = ;sec θ = ; cscθ = Copyright 0 Pearson Education, Inc.

6 Chapter Quiz s =. s = s = s = s = s = 7 7 ; 0 (c) 7 (d) 8 (e) 8 (f) 0 8. (a) F 9 F (c) F (d) 8 F (e) 8 F (f) F 87. (a) 8. ; 7 9.,,, 70.,,, ,,,,,,,, 7. ( xy=, ) ( 0.80, 0.98) 7. ( xy=, ) ( 0.98, 0.) 7. ( xy=, ) ( 0.8, ) 7. ( xy=, ) ( 0.79, 0.878) 77. quadrant I 78. quadrant IV 79. quadrant II 80. quadrant III (a). Answers will vary hr;. hr 8. (a) 0 (c) (d) (e) (f) 88. (a) (c) 0.78 (d).9 (e). (f).80 Chapter Quiz (Sections..) Copyright 0 Pearson Education, Inc.

7 8 Chapter Radian Measure and the Unit Circle... 7,00 in. 7 cm per sec Section. Linear and Angular Speed. sec. sec. (a) radians 0 cm (c) cm per sec. (a) radians cm (c) cm per sec. radians ω = θ = θ θ = radians t radian per sec 9. min 0. 9 min radian per sec radian per sec..078 radians per sec radians. 0. radians. 8 m per sec 9 7. radians per sec 8. radians per sec 9..8 radians per sec cm per sec. 8 cm. yd. sec. sec.. radian per sec 8 radian per sec 7. radian per hr radian per sec. 9. radian per min radians per min. 7 cm per min 0. mm per sec.. 8 m per min. 0 cm per min.. 00 m per min..,880 cm per min 7.. mph 8.. hr Copyright 0 Pearson Education, Inc.

8 Chapter Review Exercises 9 9. (a) radian 80 (c) 7,000 mph radian per hr 0. (a) radians per day; radian per hr 0 (c),800 km per day; km per hr (d) 8,000 km per day;00 km per hr. (a). cm per sec 0. radian per sec. larger pulley: cm per sec smaller pulley: radians per sec 8..7 cm. 9 sec.. radians per sec. ft per sec Chapter Review Exercises. radians. (a) quadrant II quadrant III (c) quadrant III (d) quadrant I. Three of the many possible answers are +, +, and n, n represents any integer in 8. in 9. in 0. mi..8 cm cm in. 7 m s. θ km 8.,000 km ;. sq units ; sq units. (a) radians in. Answers will vary, r æ ç θ ö çè ø Copyright 0 Pearson Education, Inc.

9 0 Chapter Radian Measure and the Unit Circle undefined 9. tan > tan 0. tan> sin. sin > cos. (a) A C (c) B s = sec radians. radians per sec cm. 0 cm per sec. radian per sec.. inches.. (a) 0; The face of the moon is not visible. ; Half the face of the moon is visible. (c) ; The face of the moon is completely visible. (d) ; Chapter Test (a) 8. radians 9. Half the face of the moon is visible.,000 cm 0.. undefined Copyright 0 Pearson Education, Inc.

10 Chapter Test sin = ;cos = 7 7 tan = ;csc = 7 7 sec = ;cot =. sine and cosine: (, ) ; tangent and secant: ì ï ü ís s¹ ( n+ ), where n is any integerï ý ïî ïþ ; cotangent and cosecant : { s s¹ n, where n is any integer} 7. (a) (a) radians 0 cm (c) cm per sec mi per sec 0. (a) 7 ft radian per sec Copyright 0 Pearson Education, Inc.

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

MAC Module 3 Radian Measure and Circular Functions. Rev.S08 MAC 1114 Module 3 Radian Measure and Circular Functions Learning Objectives Upon completing this module, you should be able to: 1. Convert between degrees and radians. 2. Find function values for angles