Math 1330 Precalculus Electronic Homework (EHW 3) Sections 4.1, 4.2 Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. Section 4.1 1. Section 4.1 Problem 16 6 A. B. 2 6 4 3 C. 6 D. E. 4 F. None of the above 2. In right triangle BCD with right angle C, BC= 8 and BD=17. Use the information given to find tan D and sec D. A. tan D 7 ; 15 sec D 17 8 B. tan D 15 ; 8 sec D 17 15 C. D. tan D 8 ; 15 sec D D. tan D 8 ; 15 sec D 15 17 E. tan D 24 ; 7 sec D 15 24 F. 17 15 None of the above Page 1 of 6 3. Section 4.1 Problem 36 A. sin 7 ; 9 cos 4 2 9 B. sin 9 ; 7 cos 9 2 8 C. sin 7 ; 9 cos 7 2 8 D. sin 9 2 9 ; cos 8 7 E. sin 4 2 7 ; cos 9 9 F. None of the above Section 4.1 Problem 44: Find x and csc . 4. A. x = 44; csc 44 7 B. x = 65; csc 65 7 C. x= 65 ; csc 65 7 D. x= 11 ; csc 11 7 E. x= 22 ; csc 11 7 F. None of the above Page 2 of 6 5. Triangle ABC is an equilateral triangle with AC =6; find the area of this triangle. (Hint: Draw a picture and include the altitude and mark the measures of each angle.) A. 18 3 B. 54 3 C. 18 D. 36 E. 9 3 F. None of the above 6. In triangle ABC, AC =12, the measure of angle A = 30 , and the measure of angle B = 45 . Find the area of this triangle. (Hint: Draw a picture and include the altitude from vertex C to the side AB.) A. 18 3 D. 54 1 3 B. 12 1 3 E. 54 3 C. F. 18 1 3 None of the above Page 3 of 6 Section 4.2 7. Section 4.2 Problem 12 A. 5 , 6 3 , 2 5 4 B. 7 , 6 3 , 2 5 3 C. 7 , 6 , 4 3 D. 7 , 6 3 , 2 7 4 E. 7 , 6 , 7 4 F. None of the above 8. Section 4.2 Problem 18 A. 20, 70, C. 40, 140, 636 D. 20, E. 40, 70, F. 9. 318 20, B. 318 70, 265 140, 359 None of the above Section 4.2 Problem 22 A. 4 17 B. 37 72 C. 35 72 D. 13 17 E. 13 15 F. None of the above Page 4 of 6 10. Section 4.2 Problem 38 A. 15 ft B. D. 30 ft E. 15 ft 11. 13 ft 4 C. 15 ft 2 F. None of the above Section 4.2 Problem 40 1 in 20 A. 20 in B. 20 in C. D. 125 2 in 9 E. 125 in F. None of the above 12. Section 4.2 Problem 48 A. 21 feet 4 B. 12 21 feet 4 C. 21 feet 12 D. 24 21 feet 4 E. 288 21 feet 12 F. None of the above Page 5 of 6 13. Section 4.2 Problem 60 A. 60 cm 2 B. 216 cm 2 C. 120 cm 2 D. 10 cm 2 E. 120 cm 2 F. None of the above 14. Section 4.2 Problem 62 A. 4 feet 9 B. 2 feet 3 C. 2 feet 9 D. 8 feet 3 E. 4 feet 3 F. None of the above 15. Section 4.2 Problem 68 (a) and (b). A. 8 radians per second; 44 inches per second B. 16 radians per second; 176 inches per second C. 8 radians per second; 88 inches per second D. 4 radians per second; 88 inches per second E. 4 radians per second; 176 inches per second F. None of the above Page 6 of 6 SECTION 4.1 Special Right Triangles and Trigonometric Ratios Chapter 4 Trigonometric Functions Section 4.1: Special Right Triangles and Trigonometric Ratios Special Right Triangles Trigonometric Ratios Special Right Triangles Right Triangles: MATH 1330 Precalculus 345 CHAPTER 4 Trigonometric Functions 45o-45o-90o Triangles: Theorem for 45o-45o-90o Triangles: Example: Solution: 346 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios 30o-60o-90o Triangles: Theorem for 30o-60o-90o Triangles: Example: MATH 1330 Precalculus 347 CHAPTER 4 Trigonometric Functions Solution: Additional Example 1: Solution: Part (a): Part (b): 348 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Example 2: Solution: MATH 1330 Precalculus 349 CHAPTER 4 Trigonometric Functions Additional Example 3: Solution: Part (a): Part (b): 350 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Example 4: Solution: MATH 1330 Precalculus 351 CHAPTER 4 Trigonometric Functions Additional Example 5: Solution: 352 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios Trigonometric Ratios The Three Basic Trigonometric Ratios: Example: MATH 1330 Precalculus 353 CHAPTER 4 Trigonometric Functions Solution: 354 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios The Three Reciprocal Trigonometric Ratios: MATH 1330 Precalculus 355 CHAPTER 4 Trigonometric Functions Example: Solution: 356 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Example 1: Solution: Part (a): Part (b): MATH 1330 Precalculus 357 CHAPTER 4 Trigonometric Functions Part (c): Additional Example 2: 358 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios Solution: MATH 1330 Precalculus 359 CHAPTER 4 Trigonometric Functions Additional Example 3: Solution: 360 University of Houston Department of Mathematics SECTION 4.1 Special Right Triangles and Trigonometric Ratios Additional Example 4: Solution: MATH 1330 Precalculus 361 CHAPTER 4 Trigonometric Functions 362 University of Houston Department of Mathematics Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios Answer the following. 1. 2. 9. 4 2 If two sides of a triangle are congruent, then the __________ opposite those sides are also congruent. If two angles of a triangle are congruent, then the __________ opposite those angles are also congruent. 45o x 10. 45o 3. In any triangle, the sum of the measures of its angles is _____ degrees. 4. In an isosceles right triangle, each acute angle measures _____ degrees. 5. Fill in each missing blank with one of the following: smallest, largest In any triangle, the longest side is opposite the __________ angle, and the shortest side is opposite the __________ angle. 3 2 x 11. 45o 8 x 6. Fill in each missing blank with one of the following: 30o, 60o, 90o In a 30o-60o-90o triangle, the hypotenuse is opposite the _____ angle, the shorter leg is opposite the _____ angle, and the longer leg is opposite the _____ angle. For each of the following, (a) Use the theorem for 45o-45o-90o triangles to find x. (b) Use the Pythagorean Theorem to verify the result obtained in part (a). 12. 45o 7 x 13. 9 x 7. 45o x 5 14. 12 x 8. x 15. 45o x 8 8 2 MATH 1330 Precalculus 363 Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios 22. In the figure below, an altitude is drawn to the base of an equilateral triangle. (a) Find a and b. (b) Justify the answer obtained in part (a). (c) Use the Pythagorean Theorem to find c, the length of the altitude. (Write c in simplest radical form.) 16. 45o x 2 3 17. 2 3 30o 30o 4 c 45o 60o x 60o a b 18. x For each of the following, Use the theorem for 30o-60o90o triangles to find x and y. 23. 5 2 30o x The following examples help to illustrate the theorem regarding 30o-60o-90o triangles. 19. What is the measure of each angle of an equilateral triangle? y 7 24. 22 20. An altitude is drawn to the base of the equilateral triangle drawn below. Find the measures of x and y. 30 x o y yo 25. xo 21. In the figure below, an altitude is drawn to the base of an equilateral triangle. (a) Find a and b. (b) Justify the answer obtained in part (a). (c) Use the Pythagorean Theorem to find c, the length of the altitude. x 6 3 60o y 26. 30o x y 30o 30o 10 c 60 o a 364 8 60 o b University of Houston Department of Mathematics Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios 27. x (b) Find the following: 5 60o y 28. y 60o x 34. sin A _____ sin B _____ cos A _____ cos B _____ tan A _____ tan B _____ D 25 15 3 E 29. (b) Find the following: 6 y 30. x sin D _____ sin F _____ cos D _____ cos F _____ tan D _____ tan F _____ 4 2 60o 35. Suppose that is an acute angle of a right 5 triangle and sin . Find cos and 7 tan . y 31. F (a) Use the Pythagorean Theorem to find DE. 30o x 24 y 60o 5 3 36. Suppose that is an acute angle of a right triangle and tan x cos . 32. 4 2 . Find sin and 7 37. The reciprocal of the sine function is the _______________ function. y x 30o 38. The reciprocal of the cosine function is the _______________ function. 8 Answer the following. Write answers in simplest form. 33. A 15 C 39. The reciprocal of the tangent function is the _______________ function. 40. The reciprocal of the cosecant function is the _______________ function. 17 41. The reciprocal of the secant function is the _______________ function. B 42. The reciprocal of the cotangent function is the _______________ function. (a) Use the Pythagorean Theorem to find BC. MATH 1330 Precalculus 365 Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios 47. Suppose that is an acute angle of a right 43. 6 2 10 . Find the six 3 trigonometric functions of . triangle and cot 5 x (a) Use the Pythagorean Theorem to find x. (b) Find the six trigonometric functions of . 48. Suppose that is an acute angle of a right 5 triangle and sec . Find the six 2 trigonometric functions of . (c) Find the six trigonometric functions of . 49. 44. 60o x 7 45o 3 4 (a) Use the Pythagorean Theorem to find x. (b) Find the six trigonometric functions of . (c) Find the six trigonometric functions of . 45. x 4 8 (a) Use the Pythagorean Theorem to find x. (a) Use the theorems for special right triangles to find the missing side lengths in the triangles above. (b) Using the triangles above, find the following: sin 45 _____ csc 45 _____ cos 45 _____ sec 45 _____ tan 45 _____ cot 45 _____ (c) Using the triangles above, find the following: (b) Find the six trigonometric functions of . sin 30 _____ csc 30 _____ (c) Find the six trigonometric functions of . cos 30 _____ sec 30 _____ tan 30 _____ cot 30 _____ 46. 8 6 x (a) Use the Pythagorean Theorem to find x. (b) Find the six trigonometric functions of . (c) Find the six trigonometric functions of . 366 30 2 o (d) Using the triangles above, find the following: sin 60 _____ csc 60 _____ cos 60 _____ sec 60 _____ tan 60 _____ cot 60 _____ University of Houston Department of Mathematics Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios 50. 45o 12 30o 8 60o (a) Use the theorems for special right triangles to find the missing side lengths in the triangles above. (b) Using the triangles above, find the following: sin 45 _____ csc 45 _____ cos 45 _____ sec 45 _____ tan 45 _____ cot 45 _____ (c) Using the triangles above, find the following: sin 30 _____ csc 30 _____ cos 30 _____ sec 30 _____ tan 30 _____ cot 30 _____ (d) Using the triangles above, find the following: sin 60 _____ csc 60 _____ cos 60 _____ sec 60 _____ tan 60 _____ cot 60 _____ 51. Compare the answers to parts (b), (c), and (d) in the previous two examples. What do you notice? MATH 1330 Precalculus 367 CHAPTER 4 Trigonometric Functions Section 4.2: Radians, Arc Length, and the Area of a Sector Measure of an Angle Formulas for Arc Length and Sector Area Measure of an Angle Degree Measure: 368 University of Houston Department of Mathematics SECTION 4.2 Radians, Arc Length, and the Area of a Sector Radian Measure: Example: MATH 1330 Precalculus 369 CHAPTER 4 Trigonometric Functions Solution: Converting Between Degree Measure and Radian Measure: 370 University of Houston Department of Mathematics SECTION 4.2 Radians, Arc Length, and the Area of a Sector Example: Solution: MATH 1330 Precalculus 371 CHAPTER 4 Trigonometric Functions Additional Example 1: Solution: 372 University of Houston Department of Mathematics SECTION 4.2 Radians, Arc Length, and the Area of a Sector Additional Example 2: Solution: Part (a): Part (b): Part (c): MATH 1330 Precalculus 373 CHAPTER 4 Trigonometric Functions Additional Example 3: Solution: Part (a): Part (b): Part (c): 374 University of Houston Department of Mathematics SECTION 4.2 Radians, Arc Length, and the Area of a Sector Additional Example 4: Solution: Formulas for Arc Length and Sector Area Arc Length: MATH 1330 Precalculus 375 CHAPTER 4 Trigonometric Functions Example: Solution: 376 University of Houston Department of Mathematics SECTION 4.2 Radians, Arc Length, and the Area of a Sector Area of a Sector of a Circle: Example: Solution: MATH 1330 Precalculus 377 CHAPTER 4 Trigonometric Functions Example: Solution: Part (a): 378 University of Houston Department of Mathematics SECTION 4.2 Radians, Arc Length, and the Area of a Sector Part (b): Part (c): Additional Example 1: MATH 1330 Precalculus 379 CHAPTER 4 Trigonometric Functions Solution: Additional Example 2: Solution: 380 University of Houston Department of Mathematics Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector If we use central angles of a circle to analyze angle measure, a radian is an angle for which the arc of the circle has the same length as the radius, as illustrated in the figures below. Answer the following. 7. 1 radian, since the arc length is the same as the length of the radius. Note: Standard notation is to say 1 ; radians are implied when there is no angle measure. r r 2r (a) Use this figure as a guide to sketch and estimate the number of radians in a complete revolution. 2 radians, since the arc length is twice the length of the radius. Note: Standard notation is to say 2 . r (b) Give an exact number for the number of radians in a complete revolution. Justify your answer. Then round this answer to the nearest hundredth and compare it to the result from part (a). 8. 28 cm (b) 180 _____ radians Convert the following degree measures to radians. First, give an exact result. Then round each answer to the nearest hundredth. 9. Fill in the blanks: (a) 360 _____ radians The number of radians can therefore be determined s by dividing the arc length s by the radius r, i.e. . r Find in the examples below. 1. In the figure below, 1 radian. (a) 30 (b) 90 (c) 135 10. (a) 45 (b) 60 (c) 150 11. (a) 120 (b) 225 (c) 330 12. (a) 210 (b) 270 (c) 315 13. (a) 19 (b) 40 (c) 72 14. (a) 10 (b) 53 (c) 88 7 cm 2. 30 ft 10 ft Convert the following radian measures to degrees. (b) 4 3 (c) 5 6 (b) 7 6 (c) 5 4 17. (a) (b) 11 12 (c) 61 36 9 (b) 7 18 (c) 53 30 15. (a) 3. r 0.3 m; s 72 cm 4. r 0.6 in; s 3.06 in 5. r 64 ft; s 32 ft 6. r 60 in; s 21 ft MATH 1330 Precalculus 16. (a) 18. (a) 4 2 381 Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector 19. (a) 2.5 (b) 0.506 In numbers 31-34, change to radians and then find the arc length using the formula s r . Compare results with those from exercises 23-26. 20. (a) 3.8 (b) 0.297 31. 60 ; r 12 cm Convert the following radian measures to degrees. Round answers to the nearest hundredth. 32. 90 ; r 10 in Answer the following. 21. If two angles of a triangle have radian measures 2 and , find the radian measure of the third 5 12 angle. 22. If two angles of a triangle have radian measures 3 and , find the radian measure of the third 9 8 angle. 33. 225 ; r 4 ft 34. 150 ; r 12 cm Find the missing measure in each example below. s 35. 5 6 9 in To find the length of the arc of a circle, think of the arc length as simply a fraction of the circumference of the circle. If the central angle defining the arc is given in degrees, then the arc length can be found using the formula: s 360 2 r 36. 3m 80 s Use the formula above to find the arc length s. 23. 60 ; r 12 cm 24. 90 ; r 10 in 25. 225 ; r 4 ft 26. 150 ; r 12 cm If the central angle defining the arc is instead given in radians, then the arc length can be found using the formula: s 2 2 r r Use the formula s r to find the arc length s: 27. 7 ; r 9 yd 6 28. 3 ; r 6 cm 4 29. ; r 2 ft 5 30. ; r 30 in 3 382 37. r 6 cm; 300 ; s ? 38. r 10 ft; 3 ; s ? 4 39. s 12 m; ; r ? 5 2 40. s 50 5 in; ; r ? 3 6 41. s 7 ft; 3 ; r ? 4 2 ; r ? 3 43. s 12 in; 5; r ? 42. s 20 cm; 44. s 7 in; 3; r ? 45. s 15 m; 270 ; r ? 46. s 8 yd; 225 ; r ? University of Houston Department of Mathematics Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector Answer the following. Answer the following. 47. Find the perimeter of a sector of a circle with central angle and radius 8 cm. 6 48. Find the perimeter of a sector of a circle with 7 and radius 3 ft. 4 central angle 61. A sector of a circle has central angle A 360 r 4 and area 49 cm 2 . Find the radius of the circle. 8 62. A sector of a circle has central angle area To find the area of a sector of a circle, think of the sector as simply a fraction of the circle. If the central angle defining the sector is given in degrees, then the area of the sector can be found using the formula: 5 and 6 5 2 ft . Find the radius of the circle. 27 63. A sector of a circle has central angle 120 and area 16 2 in . Find the radius of the circle. 75 2 Use the formula above to find the area of the sector: 49. 60 ; r 12 cm 50. 90 ; r 10 in 64. A sector of a circle has central angle 210 and area 21 2 m . Find the radius of the circle. 4 65. A sector of a circle has radius 6 ft and area 51. 225 ; r 4 ft 63 2 ft . Find the central angle of the sector (in 2 52. 150 ; r 12 cm radians). If the central angle defining the sector is instead given in radians, then the area of the sector can be found using the formula: A r 2 12 r 2 2 Use the formula A r to find the area of the sector: 1 2 4 cm and area 7 8 cm 2 . Find the central angle of the sector (in 49 radians). 2 53. 7 ; r 9 yd 6 54. 3 ; r 6 cm 4 55. ; r 2 ft 56. 66. A sector of a circle has radius 5 ; r 30 in 3 In numbers 57-60, change to radians and then find the area of the sector using the formula A 12 r 2 . Compare results with those from exercises 49-52. 57. 60 ; r 12 cm 58. 90 ; r 10 in 59. 225 ; r 4 ft 60. 150 ; r 12 cm MATH 1330 Precalculus Answer the following. SHOW ALL WORK involved in obtaining each answer. Give exact answers unless otherwise indicated. 67. A CD has a radius of 6 cm. If the CD's rate of turn is 900o/sec, find the following. (a) The angular speed in units of radians/sec. (b) The linear speed in units of cm/sec of a point on the outer edge of the CD. (c) The linear speed in units of cm/sec of a point halfway between the center of the CD and its outer edge. 68. Each blade of a fan has a radius of 11 inches. If the fan's rate of turn is 1440o/sec, find the following. (a) The angular speed in units of radians/sec. (b) The linear speed in units of inches/sec of a point on the outer edge of the blade. (c) The linear speed in units of inches/sec of a point on the blade 7 inches from the center. 383 Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector 69. A bicycle has wheels measuring 26 inches in diameter. If the bicycle is traveling at a rate of 20 miles per hour, find the wheels' rate of turn in revolutions per minute (rpm). Round the answer to the nearest hundredth. 70. A car has wheels measuring 16 inches in diameter. If the car is traveling at a rate of 55 miles per hour, find the wheels' rate of turn in revolutions per minute (rpm). Round the answer to the nearest hundredth. 71. A car has wheels with a 10 inch radius. If each wheel's rate of turn is 4 revolutions per second, (a) Find the angular speed in units of radians/second. (b) How fast is the car moving in units of inches/sec? (c) How fast is the car moving in miles per hour? Round the answer to the nearest hundredth. 72. A bicycle has wheels with a 12 inch radius. If each wheel's rate of turn is 2 revolutions per second, (a) Find the angular speed in units of radians/second. (b) How fast is the bicycle moving in units of inches/sec? (c) How fast is the bicycle moving in miles per hour? Round the answer to the nearest hundredth. 73. A clock has an hour hand, minute hand, and second hand that measure 4 inches, 5 inches, and 6 inches, respectively. Find the distance traveled by the tip of each hand in 20 minutes. 74. An outdoor clock has an hour hand, minute hand, and second hand that measure 12 inches, 14 inches, and 15 inches, respectively. Find the distance traveled by the tip of each hand in 45 minutes. 384 University of Houston Department of Mathematics