10-1 L E S S O N M A S T E R. Name. Vocabulary. 1. Refer to the diagram at the right. Fill in the blank. a. The leg adjacent to is.

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1 L E S S O N M S T E R Vocabular 10 Questions on SPUR Objectives 1. Refer to the diagram at the right. Fill in the blank. a. The leg adjacent to is. b. The leg opposite is. c. The hpotenuse is. C 2. Fill in the blank with the name of a trigonometric ratio. a. the of b. the of c. the of length of leg opposite length of leg adjacent to length of leg opposite length of hpotenuse length of leg adjacent to length of hpotenuse Skills Objective : pproximate values of trigonometric functions using a calculator. In 3 5, approximate each trigonometric value to the nearest thousandth. 3. Refer to the triangle at the right. a. sin b. cos c. tan 4. Refer to the triangle at the right. a. sin D b. cos D c. sin E d. cos E e. tan D f. tan E D 6.4 F E 167

2 LESSON MSTER 10 page 2 5. a. Measure the length of each side of the triangle at the right. Then estimate the trigonometric value. P i. sin R ii. cos R iii. tan R b. Measure R in the triangle at the right. Then use that measure to find each trigonometric value. i. sin R ii. cos R iii. tan R Q R 6. Refer to the regular octagon at the right. Suppose the length of each side is 12 cm. Find the length of D. C D H E G F Uses Objective F: Solve real-world problems using the trigonometr of right triangles. 7. ship sails 64 kilometers on a bearing of 20. How far east of its original position is the ship? 8. Dennis sights the top of a rocket at 54 when he stands 65 ft awa. He is 5 ft tall. bout how tall is the rocket? 5 ft U S 65 ft straight water slide makes a 40 angle with the surface of the water. If the slide is 11.5 meters high, how long is it? 168

3 L E S S O N M S T E R 10-2 Questions on SPUR Objectives Vocabular Refer to the diagram at the right. Complete the sentence with the appropriate phrase. 1. a. 1 is called a(n) 1 b. 2 is called a(n). 2. c. In geometr, 1 and 2 are called. d. How are the measures of 1 and 2 related? Skills Objective C: Determine the measure of an angle given its sine, cosine, or tangent. In 2 5, find the measure of the acute angle to the nearest degree. 2. sin cos 4. tan sin In 6 9, evaluate the functions to the nearest tenth. 6. cos tan sin cos In 10 and 11, refer to the diagram at the right. Find the measure to the nearest degree. 10. m 11. m C

4 LESSON MSTER 10-2 page 2 Uses Objective F: Solve real-world problems using the trigonometr of right triangles. 12. garage is 8 feet above the level street. The drivewa from the street to the garage is 45 feet long. Find the drivewa s angle of incline. 45 feet 8 ft 13. plane fling at 33,000 ft is 130 miles from the airport when it begins to descend. If the angle of descent is constant, find this angle. 130 miles 33,000 ft 14. If a tower 18 meters high casts a shadow 9.5 meters long, what is the angle of elevation of the sun? 18 m 15. person on top of a building finds there is a 38 angle of depression to the head of an assistant who is 170 cm tall. If the assistant is 10 meters from the building, how tall is the building? 16. The base of a 24-ft ladder is placed 8 ft from a building. a. What angle does the ladder make with the level ground? 24 ft 9.5 m m b. How high above the ground is the top of the ladder? 8 ft 170

5 L E S S O N M S T E R 10-3 Questions on SPUR Objectives Skills Objective : Find exact values of trigonometric functions of multiples of 30 or 45. In 1 9, give the exact value. 1. cos sin tan cos sin tan cos sin tan 30 In 10 13, find the trigonometric value and the indicated length. Give the exact answer x a. sin 30 a. cos 60 b. x b a 60 m a. cos 45 a. tan 60 b. a b. m Properties Objective E: Identif and use definitions and theorems relating sines, cosines, and tangents. In 14 21, fill in the blank with the measure of an acute angle. 14. sin 74 cos 15. cos 19 sin sin 45 cos 17. cos 7 sin (90 ) sin cos 23 tan 19. tan 68 sin cos (sin 88 ) 2 (cos ) (sin ) 2 (cos 14 )

6 LESSON MSTER 10-3 page 2 In 22 26, assume the angle is acute. 22. If cos x 0.49, then what is the value of sin x? If sin 3, then what is the value of cos? 24. If sin z.515, and cos z.857, what is the value of tan z? Suppose sin 5. a. What is the value of cos? b. What is the value of tan? Suppose cos 10 and tan 3, what is the value of sin? In 27 29, verif the propert for the triangle at the right. 27. (sin ) 2 (cos ) sin (90 ) cos 29. tan sin cos C 2 172

7 L E S S O N M S T E R Vocabular 10-4 Questions on SPUR Objectives 1. What is the unit circle? 2. Let (x, ) be image of (1, 0) under R. What is the relationship between (x, ) and the sine and cosine of? Skills Objective : Find exact values of trigonometric functions of multiples of 30 or Explain how to find the exact value of cos 390 without using a calculator. 4. Explain how to find the exact value of sin -300 without using a calculator. In 5 20, give the exact value. Do not use a calculator. 5. cos sin cos cos (80 ) 9. sin (-90 ) 10. cos (-90 ) 11. cos sin (-270 ) 13. cos sin sin (-330) 16. sin cos (-450) 18. sin sin sin (-300) 173

8 LESSON MSTER 10-4 page 2 Representations Objective I: Use the properties of a unit circle to find trigonometric values. In 21 26, to the nearest thousandth, find the coordinates of the image of the point (1, 0) under the given rotation. 21. R R R R R R -700 In 27 29, use the diagram of a unit circle at the right. 27. Find cos. 28. Find sin. 29. Find to the nearest degree. In 30 37, refer to the diagram at the right. Give the letter that could stand for the function value. 30. cos sin 270 (g, h) 1 (e, f) 1 (c, d) (i, j) (a, b) 1 x (.559,.829) 1 x 32. sin cos sin (-270 ) 35. cos cos sin (-278 ) Review Objective F, Lesson loading-dock ramp makes a 20 angle with the ground. If the dock is 2.5 meters high, how long is the ramp? 39. person sights the top of the San Jacinto Monument at an angle of 85 when standing 50 feet from the base of the monument. If the person is 6 feet tall, about how high is the monument? (k, l) 40. rock dropped 182 ft from the top of the Leaning Tower of Pisa lands at a point 14 ft from the base of the tower. What angle does the tower make with the ground? 174

9 L E S S O N M S T E R 10-5 Questions on SPUR Objectives Skills Objective : pproximate values of trigonometric functions using a calculator. In 1 12, use a calculator to evaluate. Round the value to the nearest thousandth. 1. cos sin cos cos sin sin cos sin (-200 ) 9. sin (-25) 10. cos sin (300) 12. cos ( ) Skills Objective : Find exact values of trigonometric functions of multiples of 30 or 45. In 13 18, true or false. Do not use a calculator. 13. sin 390 sin cos 540 cos sin -300 sin cos 210 cos sin 240 -sin cos 300 -cos 60 In 19 34, give the exact value. 19. sin cos cos sin sin cos sin cos cos sin (-45 ) 29. cos (-60 ) 30. sin (-210) 31. cos (50) 32. cos (-810 ) 33. sin (-3000) 34. cos (-585 ) 175

10 LESSON MSTER 10-5 page 2 Representations Objective I: Use the properties of a unit circle to find trigonometric values. In 35 42, for the indicated point, tell if the value 1 C for sin or cos is positive, negative, or neither. D 35., cos 36., sin 37. C, sin 38. D, cos 39. E, cos 40. F, cos 41. G, cos 42. H, sin E F G H 1 x In 43 45, refer to the unit circle at the right. Use a calculator to find the coordinates of the point to the nearest thousandth. Q P 44. Q 45. R R 16 1 x P -32 In 46 49, give the letter in the diagram at the right that could represent the given value. 46. sin sin (-270 ) 48. cos (-68 ) 49. cos 228 In 50 55, use the graph of the unit circle at the right to find the value. 50. sin 51. cos 52. cos 53. sin (e, f ) (g, h) (-.899,.438) (-.743, -.669) 1 (c, d) (a, b) 1 x (i, j) 1 1 x

11 L E S S O N M S T E R 10-6 Questions on SPUR Objectives Uses Objective G: Solve real-world problems using the Laws of Cosines. 1. Ship sights Ship at a distance of 6.4 km, and Ship sights Ship C at a distance of 7.7 km. The angle between the two sightings is 80. a. In the space below, draw and label a diagram to represent this situation. 6.4 km km C b. How far apart are Ship and Ship C? 2. Refer to the drawing at the right. t what angle should a 36-inch-wide door be opened so that distance a is at least 15 inches? 3. Refer to the drawing at the right. Maxine is designing a tent. If the two sides meet at a 40 angle, find w, the width of the tent along the ground. 4. Refer to the diagram at the right. If two planes leave erlin, one fling toward London and the other fling toward Paris, b approximatel what angle do their headings differ? London 345 km 40 8 ft 8 ft Paris w 939 km 882 km a erlin 177

12 LESSON MSTER 10-6 page 2 Representations Objective H: Find missing parts of a triangle using the Law of Cosines. 5. Find. C Find DE. E 13 D F 7. Find m G. 8. Find KJ. 15 G 6 13 I K 7 85 L 7 J H 9. Find MN. 10. Find PR. M 24.5 O P 8 Q 20 N 18 9 R 11. Find the measure of the angle. 12. Find n. Y S T U S n X 44 W T 84 U 178

13 L E S S O N M S T E R Vocabular 10-7 Questions on SPUR Objectives 1. Define triangulation. Uses Objective G: Solve real-world problems usng the Law of Sines. 2. bridge is to be built across a canon from point to point. surveor drew the diagram at the right based on measurements taken at the site. Find the length of the bridge ft C 3. In the drawing at the right, PS is the height of a mountain. Find the given measure. P a. m QRP b. m RPQ c. PR d. PS 4. s shown at the right, a ship heading due west had to detour around an oil spill. t point U, the ship steered 45 off course, and sailed until it cleared the spill. Then at point V it turned back toward its original course and intersected it at a 36 angle at point W. If the original route from U to W is 32 km long, how man additional kilometers did the ship have to sail? Q m R S W V 36 oil spill km U 5. Fire stations X and Y are 45 mi apart. The ranger at station X sees a fire at point Z such that m YXZ 30. The ranger at station Y sees the fire such that m XYZ 70. How far is the fire from each station? X Y 179

14 LESSON MSTER 10-7 page 2 Representations Objective H: Find missing parts of a triangle using the Law of Sines. 6. Find C. 7. Find DE. C E D F 8. Find KJ. K 37 J 9. Find NO. N 10 M 22 O L 10. Find GH. 11. Find TU. H G Review Objective I: Lesson 6 3 In 12 and 13, assume parabola is a translation image of parabola at the right. 12. What translation maps parabola onto parabola? Y 13. n equation for parabola is x 2. Write an equation for parabola. T 65 0 S U 10 R 10 x 0 180

15 L E S S O N M S T E R Vocabular 1. Define sine wave Questions on SPUR Objectives In 2 and 3, complete the definition. 2. If the graph of a function can be mapped onto itself under a horizontal translation of positive magnitude, then we call this tpe of function a?. 3. Situations that lead to sine waves are called?. Representations Objective J: Identif properties of the sine and cosine functions using their graphs. 4. Consider R (0,1). a. What is the first coordinate of the image? b. What is the second coordinate of the image? 5. On the grid at the right, graph the function f(x) sin x for -360 x On the grid at the right, graph the function f(x) cos x for -360 x x x 181

16 LESSON MSTER 10-8 page 2 7. Give the domain and the range of the sine function. domain range 8. Give the domain and the range of the cosine function. domain range 9. What is the -intercept of a. the sine graph? b. the cosine graph? 10. Give the least 4 nonnegative x-intercepts of a. the sine function. b. the cosine function. 11. Give the period of a. the sine function. b. the cosine function. 12. Refer to the graph at the right. a. Does this function seem to be periodic? If so, what is its period? b. Is the function graphed sinusoidal? Explain our reasoning. c. What equation might describe the graph? Review Objective C: Lesson 10-2 In 13 16, evaluate the function to the nearest tenth. 13. cos tan sin cos x 182

17 L E S S O N M S T E R 10-9 Questions on SPUR Objectives Skills Objective C: Determine the measure of an angle given its sine or cosine. In 1 4, solve for all between 0 and 180. Give to the nearest degree. 1. sin sin sin sin.988 In 5 and 6, give the exact values for all x between 0 and sin x 2 6. sin x 1 7. Suppose sin.891. Find to the nearest degree if a. is acute. b. is obtuse. Properties Objective E: Identif and use theorems relating sines and cosines. 8. Multiple choice. If sin 34 n, then (a) sin n (c) sin n? (b) sin 56 n (d) sin 146 n 9. Multiple choice. If sin 34 n, then (a) cos n (b) cos 56 n (c) cos 146 n (d) cos 56 -n 10. If is between 0 and 180, how man solutions does the equation have? a. cos.58 b. sin If sin 0.8 and 0 < < 180, give all possible values for cos. 12. If sin 0.23 and is obtuse, find cos to the nearest thousandth.? 183

18 LESSON MSTER 10-9 page 2 Uses Objective G: Solve real-world problems using the Law of Sines or the Law of Cosines. 13. In a state park, camp headquarters are 6 km from the ranger s station, and the ranger s station is 4.5 km from the park entrance. The line from the camp headquarters to the entrance forms a 48 angle with the line joining camp headquarters and the ranger s station. a. t what angle does the line joining the entrance and the camp headquarters meet the line joining the entrance and the ranger s station? (Hint: There are two possibilities.) b. Find the distance from camp headquarters to the entrance. (Give both possibilities.) Representations Objective H: Find missing parts of a triangle using the Law of Sines or the Law of Cosines. In 14 16, a triangle is described. a. Solve the triangle. b. Sketch the triangle. Give all possibilities. 14. C, with m 40, C 6, and RST, with m R 102, RS 10, and ST XYZ, with m X 72, XZ 7.3, and YZ S Z C 102 R T C Z X Y X Y

19 L E S S O N M S T E R Vocabular 1. Define radian. 100 Questions on SPUR Objectives Skills Objective : pproximate values of trigonometric functions using a calculator. In 2 9, approximate to the nearest thousandth sin ( ) 3. tan ( ) cos ( ) 5. sin (- ) tan (-2.3) 7. cos (4.6) sin 3 9. tan -5 Skills Objective : Find exact values of trigonometric functions of radian equivalents of multiples of 30 or 45. In 10 23, give the exact value. 10. cos ( ) 11. sin ( ) cos ( ) 13. sin (- ) tan (- ) 15. cos sin (- ) 17. tan ( ) sin cos sin (- ) 21. cos cos ( ) 23. sin (- )

20 LESSON MSTER 100 page 2 Skills Objective D: Convert angle measures from radians to degrees or degrees to radians. In 24 35, convert to radians In 36 47, convert to degrees Representations Objective I: Use the properties of a unit circle to find trigonometric values. In 48 55, refer to the diagram at the right. Give the letter that could represent the given function value. 48. cos 49. sin 50. sin sin cos cos sin sin (c, d) (e, f ) 1 (g, h) (a, b) 1 (i, j) x 186

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