Class: Date: Ch 9 Questions 1. Use a trigonometric ratio to determine the value of x. Round your answer to the nearest tenth. 2. 3. 4. Estimate m X to the nearest degree. 5. Katie and Matt are both flying kites on a breezy day. The string of Katie's kite is extended 864 feet, and her kite makes a 38 angle of elevation with the ground. The string of Matt's kite is extended 725 feet, and his kite makes a 40 angle of elevation with the ground. Draw a diagram that models this situation. Then determine the altitude of each kite. Round your answers to the nearest foot and show all your work. 1
6. Use ABC to answer parts (a) and (b). a. Using trigonometric ratios, name the ratio a b in two different ways. b. Using trigonometric ratios, name the ratio c a in two different ways. 7. Use ABC to answer parts (a) and (b). Show your work and round answers to the nearest tenth. a. Determine the area of the triangle. b. Determine the length of the longest side of the triangle. Post-Test Use a trigonometric ratio to determine the value of x. Round your answer to the nearest tenth. 8. 9. 10. 2
11. Use ABC to answer parts (a) and (b). a. Using trigonometric ratios, name the ratio b c in two different ways. b. Using trigonometric ratios, name the ratio b a in two different ways. 12. Use ABC to answer parts (a) and (b). Show your work and round answers to the nearest tenth. a. Determine the area of the triangle. b. Determine the length of the longest side of the triangle. 3
Mid-Chapter Test 13. In the figure, ABE and ACD are right triangles. a. How are ABE and ACD related? Justify your answer. b. Calculate the opposite-to-hypotenuse ratio for each triangle using A as the reference angle. Round answers to the nearest thousandth. c. Calculate the adjacent-to-hypotenuse ratio for each triangle using A as the reference angle. Round answers to the nearest thousandth. d. Calculate the opposite-to-hypotenuse adjacent-to-hypotenuse answers to the nearest thousandth. ratio for each triangle using A as the reference angle. Round e. What can you conclude from your results in parts (b) (d)? 14. Trevor built a wheelchair ramp in front of his grandfather s house. a. Calculate the ratio of vertical rise to horizontal run for Trevor s ramp. Write your answer as a fraction in lowest terms. b. To satisfy safety requirements, the ratio of rise to run for a wheelchair ramp cannot exceed 1 12 and the rise cannot exceed 30 inches. Does Trevor s ramp satisfy these requirements? c. Calculate the angle of incline for Trevor s ramp to the nearest tenth of a degree. Explain your reasoning. 4
15. A ski slope has an angle of elevation of 23. a. Which ratio would you use to determine the height of the ski slope? Explain your choice. b. Determine the height of the ski slope to the nearest tenth. 16. A painter leans a 12-foot ladder against a wall. The angle of elevation of the ladder is 55. 17. a. Determine the distance the ladder reaches up the wall to the nearest tenth. Determine m A to the nearest tenth of a degree. Show all your work. 18. 5
19. Use one of the triangles to calculate the exact value of each trigonometric ratio. a. sin 60 b. csc 60 c. sin 30 d. csc 30 e. sin 45 f. csc 45 20. Use a trigonometric ratio to determine the value of x. Round your answer to the nearest tenth. 21. 22. Use a trigonometric ratio to determine m A. Round your answer to the nearest tenth. 6
23. 24. A ramp at a skateboard park is 30 feet long and has a 30 incline. What is the height of the ramp? Show all your work. 25. Use a trigonometric ratio to determine the width of the rectangle shown. Round your answer to the nearest tenth of a centimeter. Show all your work. 26. Firemen are using a 72-foot ladder to reach the top of a 60-foot building. a. Calculate the distance from the bottom of the ladder to the base of the building. Round your answer to the nearest tenth. b. Use the cosine ratio to compute the measure of the angle formed where the ladder touches the top of the building. Round your answer to the nearest tenth. 7
27. Consider triangle DEF with right angle E. a. Describe the relationship between angles D and F. b. Using trigonometric ratios, name the ratio d e in two different ways. c. Using trigonometric ratios, name the ratio f d in two different ways. d. Using trigonometric ratios, name the ratio e f in two different ways. 28. Ethan has a triangular college pennant hanging in his bedroom. Calculate the area of the pennant to the nearest tenth. Show your work. 29. Use the Law of Cosines to determine x to the nearest tenth. 30. Use the Law of Sines to determine m B to the nearest tenth. 8
31. Specify whether you would use the Law of Sines or the Law of Cosines in the given situation. a. You know the lengths of all three sides of a triangle and want to solve for the measure of one of the angles. b. You know the measures of two angles and the length of the side opposite one of them and want to solve for the length of the side opposite the other given angle. c. You know the lengths of two sides of a triangle and the measure of the included angle and want to solve for the length of the third side. Standardized Test Practice 32. Which of the following is equivalent to csc x? a. 1 sec x b. 1 tanx c. (sin x)(cos x) d. 1 sinx 33. Eric is flying an airplane at an altitude of 2200 feet. He sees his house on the ground at a 45 angle of depression. What is Eric s horizontal distance from his house at this point? a. 110 feet b. 220 feet c. 1100 feet d. 2200 feet 9
34. What is the length of AB to the nearest tenth? a. 1.8 meters b. 3.8 meters c. 4.0 meters d. 4.1 meters 35. Which is closest to the value of the adjacent-to-hypotenuse ratio for a 45 angle? a. 1.414 b. 0.5 c. 1 d. 0.707 36. What is m N to the nearest degree? a. 16 b. 17 c. 63 d. 73 10
37. What is the area of this triangular garden plot to the nearest tenth? a. 428.5 square feet b. 225.0 square feet c. 214.0 square feet d. 69.5 square feet 38. Luis is standing on a street in New York City looking at the top of the Empire State Building with a 30 angle of elevation. He is 767.6 meters from the Empire State Building. How tall is the Empire State Building? a. 383.8 meters b. 443.2 meters c. 664.8 meters d. 1329.5 meters 39. Which one of the following statements is true? a. As the measure of an acute angle increases, the sine and cosine of the angle increase. b. As the measure of an acute angle increases, the sine and cosine of the angle decrease. c. As the measure of an acute angle increases, the sine and tangent of the angle increase. d. As the measure of an acute angle increases, the cosine and tangent of the angle decrease. 11
40. In the figure shown, cos P = 0.60. What is the length of PN? a. 0.25 centimeter b. 14.4 centimeters c. 40 centimeters d. 44 centimeters 41. A proposed wheelchair ramp is shown. What is the rise of the ramp to the nearest inch? a. 14 inches b. 12 inches c. 81 inches d. 181 inches 42. Which ratio has the same value as sec E? a. cos E b. cot E c. cos G d. csc G 12
43. In the diagram shown, m B = 42 and AB = 25 feet. Which equation can be used to calculate the value of x? a. x = 25(sin 42 ) b. x = 25(cos 42 ) c. x = 25(tan 42 ) d. sin 42 x = 25 44. Which is NOT a valid conclusion you can draw about this figure? a. AEC BDC b. slope of AC = slope of BC c. AE AC = BD BC d. AE AC = BD DC 45. Which could you use to calculate the length of RT? a. Triangle Area Formula b. Pythagorean Theorem c. Law of Sines d. Law of Cosines 13
46. Which of the following can be used to determine m A? Ê a. sin 1 BC ˆ Ë Á AC Ê b. cos 1 BC ˆ Ë Á AC Ê c. sin 1 AC ˆ Ë Á BC AC d. BC 47. In the diagram shown, a 12-foot slide is attached to a swing set. The slide makes a 65 angle with the swing set. Which answer most closely represents the height of the slide? a. 5.0 feet b. 5.6 feet c. 10.9 feet d. 25.7 feet 48. What is the value of csc x, if sinx = 5 13? a. b. c. d. 12 13 13 12 13 5 12 5 14
49. If cos A 0.67, which of the following statements must be true? a. The measure of A is between 30 and 45. b. The measure of A is between 45 and 60. c. The measure of A is between 60 and 75. d. The measure of A is between 75 and 90. 50. Which of the following statements is NOT true? a. The cosine of an acute angle is always less than or equal to one. b. The sine of an acute angle is always less than or equal to one. c. The tangent of an acute angle is always less than or equal to one. d. The value of the sine of an angle divided by the value of the cosine of the angle is equal to the value of the tangent of the angle. 51. Which is the exact value of cot 60? a. b. c. d. 1 3 3 1 2 3 3 2 15
Three Angle Measure Introduction to Trigonometry 52. Analyze triangle ABC and triangle DEF. Use A and D as the reference angles. a. Identify the leg opposite A, the leg adjacent to A, and the hypotenuse in ABC. b. Calculate the length of the hypotenuse of triangle ABC. Round your answer to the nearest tenth. opposite c. Calculate the ratios hypotenuse, adjacent opposite, and hypotenuse adjacent Round your answers to the nearest thousandth if necessary. for the reference angle in triangle ABC. d. Describe the relationship between ABC and DEF. Explain your reasoning. e. Calculate the length of the hypotenuse in DEF without using the Pythagorean Theorem. Explain your reasoning. opposite f. Calculate the ratios hypotenuse, adjacent opposite, and hypotenuse adjacent Round your answers to the nearest thousandth if necessary. for the reference angle in DEF. g. Compare the values of the three ratios for ABC and DEF. What do you observe? Why do you think this is true? The Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent Use the tangent ratio, the cotangent ratio, or the inverse tangent to solve for x. Round each answer to the nearest tenth. 53. 16
54. The Sine Ratio Sine Ratio, Cosecant Ratio, and Inverse Sine Use the sine ratio, the cosecant ratio, or the inverse sine to solve for x. Round each answer to the nearest tenth. 55. 56. 57. 58. 17
59. A roof truss is shown in the following figure. Use the figure to complete parts (a) through (d). Round each answer to the nearest hundredth. 60. a. Determine the height CG of the roof truss. b. Determine AB. c. Determine the measure of angle BGC. d. Determine the length BG of the support beam. The Cosine Ratio Cosine Ratio, Secant Ratio, and Inverse Cosine Use the cosine ratio, the secant ratio, or the inverse cosine to solve for x. Round each answer to the nearest tenth. 61. 62. 18
63. A bridge is shown in the following figure. Use the figure and the fact that AGC is congruent to EGC to complete parts (a) through (e). Round each answer to the nearest tenth. a. Determine the width AE of the bridge. b. Determine the height CG of the bridge. c. Determine CH. d. Determine the measure of BHC. e. Does CH bisect ACG? Explain your reasoning. We Complement Each Other! Complement Angle Relationships 64. A pilot and co-pilot are performing a test run in a new airplane. The pilot is required to take off and fly in a straight path at an angle of elevation that is between 33 and 35 degrees until the plane reaches an altitude of 10,000 feet. When the plane reaches 10,000 feet, the co-pilot will take over. a. Draw a figure to model this situation. Label the angle of elevation and the side opposite the angle of elevation. Label the side adjacent to the angle of elevation as x and the hypotenuse as y. b. Determine the minimum and maximum horizontal distance between the point of take-off and the point at which the co-pilot takes over. Round each distance to the nearest tenth. c. What is the minimum distance that the pilot flies the plane? What is the maximum distance that the pilot flies the plane? Round each distance to the nearest tenth. 19
Time to Derive! Deriving the Triangle Area Formula, the Law of Sines, and the Law of Cosines 65. Emily and Joe are designing a fenced backyard play space for their children Max and Caroline. They start out by considering two designs for a triangular play space. They have made measurements in their yard and determined that either design would fit into the space that is available. mily s Design Joe s Design a. Explain how Emily and Joe can use trigonometry to calculate the area and perimeter of the possible play spaces. b. Calculate the area of the play space for each design. c. Calculate the perimeter of the play space for each design. d. Which design do you think Emily and Joe should choose? Explain your reasoning. 66. Emily s brother-in-law Chris is an architect. She has asked him to design the placement of the playground equipment in her children s new play space. He sent her a diagram of the play space with the measurements shown. a. Explain how Emily can calculate the area and perimeter of the play space in Chris s design. b. Calculate the area of the play space for Chris s design. c. Calculate the perimeter of the play space for Chris s design. 20
Three Angle Measure Introduction to Trigonometry Vocabulary Use the diagram to complete each sentence. 67. If b is the opposite side, then x is the. 68. If y is the reference angle, then b is the. 69. If x is the reference angle, then b is the. Problem Set opposite Determine the ratio hypotenuse fractions in simplest form. using A as the reference angle in each triangle. Write your answers as 70. 71. 21
72. 73. 74. adjacent Determine the ratio hypotenuse fractions in simplest form. using A as the reference angle in each triangle. Write your answers as 75. 22
76. 77. 78. 79. 80. 23
opposite Determine the ratios hypotenuse, adjacent opposite, and hypotenuse adjacent triangle. Write your answers as fractions in simplest form. using A as the reference angle in each 81. 82. 83. 84. 24
In each figure, triangles ABC and DEF are similar by the AA Similarity Theorem. Calculate the indicated ratio twice, first using ABC and then using ADE. 85. opposite hypotenuse for reference angle A 86. adjacent hypotenuse for reference angle A 87. opposite hypotenuse for reference angle A 25
88. adjacent hypotenuse for reference angle A 89. opposite for reference angle A adjacent 90. opposite for reference angle A adjacent 26
The Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent Vocabulary Match each description to its corresponding term for triangle EFG. a. tangent b. cotangent c. inverse tangent 91. EG EF in relation to G 92. EF EG Ê 93. tan 1 Ë Á 94. in relation to G EF EG ˆ Problem Set in relation to G Calculate the tangent of the indicated angle in each triangle. Write your answers in simplest form. 95. tan B = tan B = 27
96. tan C = 97. tan C = 98. tan D = 99. tan D = 28
Calculate the cotangent of the indicated angle in each triangle. Write your answers in simplest form. 100. cot A = 101. cot A = 102. cot F = 103. cot F = 29
104. cot A = Use a calculator to approximate each tangent ratio. Round your answers to the nearest hundredth. 105. tan 60 106. tan 15 107. tan 89 Use a calculator to approximate each cotangent ratio. Round your answers to the nearest hundredth. 108. cot 60 109. cot 15 110. cot 45 111. cot 75 112. cot 10 113. cot 30 114. Use a tangent ratio or a cotangent ratio to calculate the missing length of each triangle. Round your answers to the nearest hundredth. 30
115. 116. 117. 118. Calculate the measure of angle X for each triangle. Round your answers to the nearest hundredth. 119. 31
120. 121. 122. 123. Solve each problem. Round your answers to the nearest hundredth. 124. A boat travels in the following path. How far north did it travel? 32
125. A surveyor makes the following diagram of a hill. What is the height of the hill? 126. To calculate the height of a tree, a botanist makes the following diagram. What is the height of the tree? 127. A moving truck is equipped with a ramp that extends from the back of the truck to the ground. When the ramp is fully extended, it touches the ground 12 feet from the back of the truck. The height of the ramp is 2.5 feet. Calculate the measure of the angle formed by the ramp and the ground. 128. A park has a skateboard ramp with a length of 14.2 feet and a length along the ground of 12.9 feet. The height is 5.9 feet. Calculate the measure of the angle formed by the ramp and the ground. 33
The Sine Ratio Sine Ratio, Cosecant Ratio, and Inverse Sine Vocabulary Write the term from the box that best completes each statement. sine cosecant inverse sine 129. The of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of a side that is opposite the angle. 130. The of x is the measure of an acute angle whose sine is x. 131. The of an acute angle in a right triangle is the ratio of the length of the side that is opposite the angle to the length of the hypotenuse. 132. Problem Set Calculate the sine of the indicated angle in each triangle. Write your answers in simplest form. 133. sin B = sin C = 34
134. sin C = 135. sin D = Calculate the cosecant of the indicated angle in each triangle. Write your answers in simplest form. 136. csc A = 35
137. csc A = 138. csc F = 139. csc F = 140. csc P = Use a calculator to approximate each sine ratio. Round your answers to the nearest hundredth. 141. sin 30 142. sin 45 143. sin 60 144. sin 15 36
145. sin 75 146. sin 5 Use a calculator to approximate each cosecant ratio. Round your answers to the nearest hundredth. 147. csc 90 148. csc 120 149. csc 30 150. csc 15 151. csc 60 152. Use a sine ratio or a cosecant ratio to calculate the missing length of each triangle. Round your answers to the nearest hundredth. 153. 154. 155. 37
156. Calculate the measure of angle X for each triangle. Round your answers to the nearest hundredth. 157. 158. 159. 160. 38
Solve each problem. Round your answers to the nearest hundredth. 161. A scout troop traveled 12 miles from camp, as shown on the map below. How far north did they travel? 162. An ornithologist tracked a Cooper s hawk that traveled 23 miles. How far east did the bird travel? 163. An architect needs to use a diagonal support in an arch. Her company drew the following diagram. How long does the diagonal support have to be? 39
164. A hot air balloon lifts 125 feet into the air. The diagram below shows that the hot air balloon was blown to the side. How long is the piece of rope that connects the balloon to the ground? 165. Jerome is flying a kite on the beach. The kite is attached to a 100-foot string and is flying 45 feet above the ground, as shown in the diagram. Calculate the measure of the angle formed by the string and the ground. 166. An airplane ramp is 58 feet long and reaches the cockpit entrance 19 feet above the ground, as shown in the diagram. Calculate the measure of the angle formed by the ramp and the ground. 167. Bleachers in a stadium are 4 meters tall and have a length of 12 meters, as shown in the diagram. Calculate the measure of the angle formed by the bleachers and the ground. 40
168. A 20-foot flagpole is raised by a 24-foot rope, as shown in the diagram. Calculate the measure of the angle formed by the rope and the ground. Define the term in your own words. 169. inverse cosine Problem Set Calculate the cosine of the indicated angle in each triangle. Write your answers in simplest form. 170. 171. cos B = 172. cos B = cos C = 41
173. cos C = 174. cos D = 175. Calculate the secant of the indicated angle in each triangle. Write your answers in simplest form. 176. sec A = 177. sec F = 42
178. sec F = 179. sec P = 180. sec P = Use a calculator to approximate each cosine ratio. Round your answers to the nearest hundredth. 181. cos 30 182. cos 45 183. cos 60 184. cos 15 185. cos 75 Use a calculator to approximate each secant ratio. Round your answers to the nearest hundredth. 186. sec 45 187. sec 25 188. sec 75 189. sec 60 43
Use a cosine ratio or a secant ratio to calculate the missing length of each triangle. Round your answers to the nearest hundredth. 190. 191. 192. 193. 194. Calculate the measure of angle X for each triangle. Round your answers to the nearest hundredth. 195. 44
196. 197. 198. 199. Solve each problem. Round your answers to the nearest hundredth. 200. The path of a model rocket is shown below. How far east did the rocket travel? 45
201. An ichthyologist tags a shark and charts its path. Examine his chart below. How far south did the shark travel? 202. A kite is flying 120 feet away from the base of its string, as shown below. How much string is let out? 203. A pole has a rope tied to its top and to a stake 15 feet from the base. What is the length of the rope? 204. A ladder is leaning against the side of a house, as shown in the diagram. The ladder is 24 feet long and makes a 76 angle with the ground. How far from the edge of the house is the base of the ladder? 205. A rectangular garden 9 yards long has a diagonal path going through it, as shown in the diagram. The path makes a 34 angle with the longer side of the garden. Determine the length of the path. 46
We Complement Each Other! Complement Angle Relationships Problem Set For each right triangle, name the given ratio in two different ways. 206. a c 207. d e 208. p m 209. s r 47
210. w v Determine the trigonometric ratio that you would use to solve for x in each triangle. Explain your reasoning. You do not need to solve for x. 211. 212. 213. 214. 48
215. Solve each problem. Round your answers to the nearest hundredth. 216. A surveyor is 3 miles from a mountain. The angle of elevation from the ground to the top of the mountain is 15. What is the height of the mountain? 217. The angle of elevation from a ship to a 135-foot-tall lighthouse is 2. How far is the ship from the lighthouse? 218. The Statue of Liberty is about 151 feet tall. If the angle of elevation from a tree in Liberty State Park to the statue s top is 1.5, how far is the tree from the statue? 219. A plane is spotted above a hill that is 12,000 feet away. The angle of elevation to the plane is 28. How high is the plane? 220. During the construction of a house, a 6-foot-long board is used to support a wall. The board has an angle of elevation from the ground to the wall of 67. How far is the base of the wall from the board? 221. Museums use metal rods to position the bones of dinosaurs. If an angled rod needs to be placed 1.3 meters away from a bone, with an angle of elevation from the ground of 51, what must the length of the rod be? Solve each problem. Round your answers to the nearest hundredth. 222. The angle of depression from the top of a building to a telephone line is 34. If the building is 25 feet tall, how far from the building does the telephone line reach the ground? 223. An airplane flying 3500 feet from the ground sees an airport at an angle of depression of 77. How far is the airplane from the airport? 224. To determine the depth of a well s water, a hydrologist measures the diameter of the well to be 3 feet. She then uses a flashlight to point down to the water on the other side of the well. The flashlight makes an angle of depression of 79. What is the depth of the well water? 225. A zip wire from a tree to the ground has an angle of depression of 18. If the zip wire ends 250 feet from the base of the tree, how far up the tree does the zip wire start? 226. From a 50-foot-tall lookout tower, a park ranger sees a fire at an angle of depression of 1.6. How far is the fire from the tower? 227. The Empire State Building is 448 meters tall. The angle of depression from the top of the Empire State Building to the base of the UN building is 74. How far is the UN building from the Empire State Building? 49
228. A factory conveyor has an angle of depression of 18 and drops 10 feet. How long is the conveyor? 229. A bicycle race organizer needs to put up barriers along a hill. The hill is 300 feet tall and from the top makes an angle of depression of 26. How long does the barrier need to be? Time to Derive! Deriving the Triangle Area Formula, the Law of Sines, and the Law of Cosines Vocabulary Define each term in your own words. 230. Law of Sines 231. Problem Set Determine the area of each triangle. Round your answers to the nearest tenth. 232. 233. 50
234. 235. 236. Determine the unknown side length x by using the Law of Sines. Round your answers to the nearest tenth. 237. 238. 239. 51
240. 241. Determine m B by using the Law of Sines. Round your answers to the nearest tenth. 242. 243. 52
244. 245. 246. Determine the unknown side length by using the Law of Cosines. Round your answers to the nearest tenth. 247. 248. 53
249. 250. 54
Ch 9 Questions Answer Section 1. sin48 = x 20 20(sin48 ) = x 14.9 mm x 2. cos 39 = 12 x x = 12 cos 39 x 15.4in. 3. tan 64 = x 15 15(tan64 ) = x 30.8ft x Ê 4. m X =tan 1 5ˆ Ë Á 8 32 5. sin38 = altitude 864 altitude = 864sin38 altitude 532 Katie' s kite is about532feet above the ground. sin40 = altitude 725 altitude = 725sin40 altitude 466 Matt' s kite is about466feet above the ground. 6. a. tan A= a b andcot B = a b b. sec B = c a andcsc A = c a 1
7. a. A = 1 ab sinc 2 A = 1 2 (20)(13)(sin110 ) A 122.2 The area of the triangle is approximately 122.2 square centimeters. b. I used the Law of Cosines. c 2 = a 2 + b 2 2ab cos C c 2 = 20 2 + 13 2 2(20)(13)(cos 110 ) c 2 746.850 c 27.3 8. sin26 = x 8 8(sin26 ) = x 3.5mm x 9. cos 58 = 40 x x = The length of the longest side of the triangle is approximately 27.3 centimeters. 40 cos 58 x 75.5 in. 10. tan71 = x 27 27(tan71 ) = x 78.4 ft x 11. a. sin B = b c and cos A = b c b. tan B = b a and cot A = b a 2
12. a. A = 1 ab sinc 2 A = 1 2 (10)(15)(sin78 ) A 73.4 The area of the triangle is approximately 73.4 square inches. b. I used the Law of Cosines to calculate the length of the third side in order to determine which side is the longest. c 2 = a 2 + b 2 2ab cos C c 2 = 10 2 + 15 2 2(10)(15)(cos 78 ) c 2 262.626 c 16.2 The length of the longest side of the triangle is approximately 16.2 inches. 13. a. The two triangles are similar. A is common to both triangles and each has a right angle, so ABE ACD by the AA Similarity Theorem. I could also use the SSS Similarity Theorem or the SAS Similarity Theorem. b. In ABE, the ratio is 5.0 11.8 0.424. In ACD, AD = 11.8cm + 9.5cm = 21.3cm, so the ratio is 9.0 21.3 0.423. c. In ABE, the ratio is 10.7 11.8 0.907. In ACD, AC = 10.7cm + 8.6cm = 19.3cm, so the ratio is 19.3 21.3 0.906. d. In ABE, the ratio is 0.424 0.907 0.467. In ACD, the ratio is 0.423 0.906 0.467. e. The two ratios in each question are equal. 3
14. a. The ratio is 2.5 37.5 = 1 15. b. Yes, Trevor s ramp satisfies the requirements because 1 15 < 1 and 2.5 feet does not exceed 30 12 inches. Two and five-tenths feet is equal to 30 inches: 2.5 12 = 30. c. The angle of incline is approximately 3.8. I knew the lengths of the two legs of the right triangle, so I used the inverse tangent to calculate the measure of the angle of incline. Ê m A = tan 1 2.5 ˆ Ë Á 37.5 m A 3.8 15. a. I would use the tangent ratio because I know the measure of an acute angle of a right triangle and the length of the adjacent side, and I need to calculate the length of the opposite side. b. The height of the ski slope is approximately 1305.7 feet. tan23 = x 3076 3076(tan23 ) = x 1035.7 x 16. a. The ladder reaches approximately 9.8 feet up the wall. sin55 = x 12 12(sin55 ) = x 9.8 x 17. tan A = 28 13 Ê m A = tan 1 28ˆ Ë Á 13 65.1 18. cos A = 25 45 Ê m A = cos 1 25ˆ Ë Á 45 56.3 4
19. a. sin60 = 9 3 18 = 3 2 b. csc 60 = 18 9 3 = 2 3 c. sin30 = 9 18 = 1 2 d. csc 30 = 18 9 = 2 e. sin45 = 10 10 2 = 1 2 f. csc 45 = 10 2 10 20. sin55 = 27 x x = 27 sin55 x 33.0 ft = 2 21. tan73 = 8 x x = 8 tan73 x 2.4 in. Ê 22. m A = tan 1 5ˆ Ë Á 4 51.3 Ê 23. m A = cos 1 10ˆ Ë Á 11 24.6 24. sin30 = x 30 30(sin30 ) = x 15 = x The height of the ramp is 15 feet. 25. cos 59 = x 37 37(cos 59 ) = x 19.1 x The width of the rectangle is about 19.1 centimeters. 5
26. a. 60 2 + b 2 = 72 2 b 2 = 72 2 60 2 b 2 = 5184 3600 b 2 = 1584 b = 1584 39.8 The distance from the bottom of the ladder to the base of the building is approximately 39.8 feet. b. cos B = 60 72 Ê m B= cos 1 60ˆ Ë Á 72 33.6 The measure of the angle formed where the ladder touches the top of the building is approximately 33.6. 27. a. Angles D and F are complementary. b. sin D = d e and cos F = d e c. tan F = f d and cot D = f d d. sec D = e f and csc F = e f 28. A = 1 ab (sinc) 2 A = 1 2 (15)(15)(sin28 ) A 52.8 The area of the pennant is approximately 52.8 square inches. 29. c 2 = a 2 + b 2 2ab cos C x 2 = 8 2 + 12 2 2(8)(12)(cos 130 ) x 2 = 64 + 144 192cos 130 x 2 331.415 x 18.2 6
30. sina a sin75 15 = sinb b = sinb 13 13sin75 = 15 sinb sinb = 13sin75 15 0.837 m B 56.8 31. a. I would use the Law of Cosines. b. I would use the Law of Sines. c. I would use the Law of Cosines. 32. D 33. D 34. B 35. D 36. B 37. C 38. B 39. C 40. C 41. A 42. D 43. A 44. D 45. C 46. A 47. A 48. C 49. B 50. C 51. A 7
52. a. The leg opposite A is BC. The leg adjacent to A is AC. The hypotenuse is AB. b. c 2 = a 2 + b 2 c 2 = 7 2 + 10 2 c 2 = 49 + 100 c 2 = 149 c = 149 c 12.2 The length of the hypotenuse is approximately 12.2 centimeters. c. opposite hypotenuse = 7.0 12.2 0.574 adjacent hypotenuse = 10.0 12.2 0.820 opposite adjacent = 0.574 0.820 = 0.7 d. The two triangles are similar by the AA Similarity Theorem because m A = m D = 35 and m C = m F = 90. e. Corresponding sides of similar triangles are proportional, so I can write and solve a proportion. DF AC = DE AB 15.0 10.0 = DE 12.2 DE = ( 15.0) ( 12.2) 100 = 18.3 The length of the hypotenuse of DEF is 18.3 centimeters. f. opposite hypotenuse = 10.5 18.3 0.574 adjacent hypotenuse = 15.0 18.3 0.820 opposite adjacent = 0.574 0.820 = 0.7 8
g. All three ratios are equal for the two triangles. This is true because the two right triangles are similar, and corresponding sides of similar triangles are proportional. 53. cot 72 = x 12 12cot 72 = x x 3.9 in. Ê 54. x = tan 1 21ˆ Ë Á 23 x 42.4 55. csc 57 = x 11 11csc 57 = x x 13.1ft Ê 56. x = sin 1 17ˆ Ë Á 25 x 42.8 Ê 57. x = sin 1 20ˆ Ë Á 58 x 20.2 58. sin50 = x 75 75sin50 = x x 57.5m 9
59. a. sin40 = CG 16 16sin40 = CG 10.28 CG The height of the roof truss is about 10.28 feet. b. csc 40 = AB 6 6csc 40 = AB 9.33 AB The length of AB is about 9.33 feet. Ê c. m BGC = sin 1 16 9.33ˆ Ë Á 10.28 Ê m BGC = sin 1 6.67 ˆ Ë Á 10.28 m BGC 40.45 The measure of angle BGC is about 40.45 degrees. d. (CG) 2 = (BG) 2 + (BC) 2 60. cos 33 = x 15 15cos 33 = x (10.28) 2 = (BG) 2 + (6.67) 2 (BG) 2 = 61.1895 BG 7.82 x 12.6m Ê 61. x = cos 1 12ˆ Ë Á 19 x 50.8 Ê 62. x = cos 1 17ˆ Ë Á 22 x 39.4 The length of the support beam is about 7.82 feet. 10
63. a. cos 36 = AG 40 40cos 36 = AG 32.4 AG AE 2 32.4 64.8 The width of the bridge is about 64.8 feet. b. (AC) 2 = (AG) 2 + (CG) 2 (40) 2 = (32.4) 2 + (CG) 2 (CG) 2 = 550.24 CG 23.5 The height of the bridge is about 23.5 feet. c. sec 27 = CH 23.5 23.5sec 27 = CH 26.4 CH The length of CH is about 26.4 feet. Ê d. m BHC = cos 1 12 ˆ Ë Á 26.4 m BHC 63.0 The measure of BHC is about 63 degrees. e. Yes.CH bisects ACG. The measure of BHC is 63 degrees. By the Triangle Sum Theorem, the measure of BCH is 27 degrees. Because BCH and HCG are congruent, CH bisects ACG. 11
64. a. b. Minimum Maximum tan35 = 10,000 tan33 = 10,000 x x x tan35 = 10,000 x = 10,000 tan35 x 14,281.5 x tan33 = 10,000 x = 10,000 tan33 x 15,398.6 The minimum horizontal distance between the point of take-off and the point at which the co-pilot takes over is approximately 14,281.5 feet. The maximum horizontal distance between the point of take-off and the point at which the co-pilot takes over is approximately 15,398.6 feet. c. Minimum Maximum sin35 = 10,000 sin33 = 10,000 y y y sin35 = 10,000 y = 10,000 sin35 y 17,434.5 y sin33 = 10,000 y = 10,000 sin33 y 18,360.8 The minimum distance that the pilot flies the plane is approximately 17,434.5 feet. The maximum distance that the pilot flies the plane is approximately 18,360.8 feet. 12
65. a. For both designs, they know the lengths of two sides of the triangle and the measure of the included angle. They can calculate the area of each triangle by using the formula for the area of any triangle. They can calculate the perimeter of each triangle by using the Law of Cosines to calculate the length of the third side and then adding the lengths of the three sides. b. for Emily s design: A = 1 ac sinb 2 = 1 2 (11)(8)(sin80 ) 43.3 The area for Emily s design is approximately 43.3 square feet. for Joe s design: A = 1 ac sinb 2 = 1 2 (11)(8)(sin110 ) 41.3 The area for Joe s design is approximately 41.3 square feet. c. for Emily s design: for Emily s design: b 2 = a 2 + c 2 2ac cos B b 2 = a 2 + c 2 2ac cos B b 2 = 11 2 + 8 2 2(11)(8) cos 80 b 2 = 121 + 64 176cos 80 154.4 b = 154.4 b 12.4 b 2 = 11 2 + 8 2 2(11)(8) cos 110 b 2 = 121 + 64 176cos 110 245.2 b = 245.2 b 15.7 For Emily s design, the perimeter of the play space is approximately 11 + 8 + 12.4, or 31.4 feet. For Joe s design, the perimeter is approximately 11 + 8 + 15.7, or 34.7 feet. d. Answers will vary. A sample answer is given. I think they should choose Emily s design because the triangle has a greater area but smaller perimeter than the triangle in Joe s design. Her design gives the children a larger area in which to play and requires less fencing, which will save money when they buy the fencing. 13
66. a. First Emily can use the Triangle Sum Theorem to determine m B. Then she can use the Law of Sines to calculate a. After these two steps, she will have the lengths of two sides and the measure of the included angle, so she can calculate the area and perimeter. b. m B = 180 85 42 = 53 sina a sin 85 a = sinb b = sin63 11 11sin85 = a sin63 a = 11sin85 sin53 a 13.7cm A = 1 ab sinc 2 = 1 2 (13.7)(11)(sin42 ) 50.4 For Chris s design, the area of the play space is approximately 50.4 square feet. c. c 2 = a 2 + b 2 2ab cos C c 2 = 13.7 2 + 11 2 2(13.7)(11) cos 42 c 2 = 187.69 + 121 301.4cos 42 84.7 c = 84.7 9.2 P = 11 + 13.7 + 9.2 = 33.9 For Chris s design, the perimeter of the play space is approximately 33.9 feet. 67. reference angle 68. adjacent side 69. opposite side opposite 70. hypotenuse = 6 10 = 3 5 71. opposite hypotenuse = 24 26 = 12 13 14
72. c 2 = a 2 + b 2 c 2 = 15 2 + 8 2 c 2 = 225 + 64 = 289 c = 289 = 17 opposite hypotenuse = 15 17 73. c 2 = a 2 + b 2 c 2 = 7 2 + 24 2 c 2 = 49 + 576 = 625 c = 625 = 25 opposite hypotenuse = 7 25 74. c 2 = a 2 + b 2 75. 76. c 2 = 1 2 Ê ˆ + 3 Ë Á c 2 = 1 + 3 = 4 c = 4 = 2 opposite hypotenuse = 1 2 adjacent hypotenuse = 20 25 = 4 5 adjacent hypotenuse = 16 34 = 8 17 77. c 2 = a 2 + b 2 2 c 2 = 1.4 2 + 4.8 2 c 2 = 1.96 + 23.04 = 25.0 c = 25.0 = 5.0 adjacent hypotenuse = 4.8 5.0 = 24 25 15
78. c 2 = a 2 + b 2 c 2 = 4 2 + 4 2 c 2 = 16 + 16 = 32 c = 32 = 4 2 adjacent hypotenuse = 4 4 2 = 1 2 or 2 2 79. c 2 = a 2 + b 2 c 2 = 2.4 2 + 1.0 2 c 2 = 1.00 + 5.76 = 6.76 c = 6.76 = 2.6 adjacent hypotenuse = 1.0 2.6 = 5 13 80. c 2 = a 2 + b 2 c 2 = 2 2 + (2 3) 2 c 2 = 4 + 12 = 16 81. c = 16 = 4 adjacent hypotenuse = 2 3 4 = opposite hypotenuse = 18 30 = 3 5 adjacent hypotenuse = 24 30 = 4 5 opposite adjacent = 18 24 = 3 4 3 2 16
82. b 2 = c 2 a 2 b 2 = 51 2 24 2 b 2 = 2601 576 = 2025 b = 2025 = 45 opposite hypotenuse = 24 51 = 8 17 adjacent hypotenuse = 45 51 = 15 17 opposite adjacent = 24 45 = 8 15 83. a 2 = c 2 b 2 a 2 = 29 2 20 2 a 2 = 841 400 = 441 a = 441 = 21 opposite hypotenuse = 21 29 adjacent hypotenuse = 20 29 opposite adjacent = 21 20 84. a 2 = c 2 b 2 b 2 = (5 2) 2 5 2 b 2 = 50 25 = 25 b = 25 = 5 opposite hypotenuse = 5 2 or 5 2 2 adjacent hypotenuse = 5 2 or 5 2 2 opposite adjacent = 5 5 = 1 17
85. AE = 4 + 4 = 8 AD = 5 + 5 = 10 opposite In ABC, hypotenuse = 3 5. In ADE, 86. AE = 15 + 30 = 45 opposite hypotenuse = 6 10 = 3 5. AD = 17 + 34 = 51 adjacent In ABC, hypotenuse = 15 17. In ADE, 87. AE = 10 + 15 = 25 adjacent hypotenuse = 45 51 = 15 17. AD = 10 2 + 15 2 = 25 2 In In ABC, ADE, 88. AE = 2 3 + 3 = 3 3 AD = 4 + 2 = 6 In In ADE, ADE, opposite hypotenuse = 10 10 2 = 1 2 or 2 2. opposite hypotenuse = 25 25 2 = 1 2 or 2 2. adjacent hypotenuse = 2 3 4 adjacent hypotenuse = 3 3 6 89. In ABC, opposite adjacent = 8 15. In ADE, opposite adjacent = 24 45 = 8 15. 90. In ABC, opposite adjacent = 2.1 2.0 = 21 20. In 91. B 92. A 93. C 94. tan B = 2 2 = 1 ADE, opposite adjacent = 8.4 8.0 = 21 20. = = 3 2. 3 2. 18
95. tan B = 3 2 3 2 = 1 96. tan C = 25 20 = 5 4 97. tan C = 32 40 = 4 5 98. tan D = 2 2 15 99. tan D = 100. cot A = 4 3 3 5 5 = 3 5 25 101. cot A = 6 8 = 3 4 102. cot F = 7 15 103. cot F = 2 6 = 6 104. cot A = 32 40 = 4 5 105. 1.73 106. 0.27 107. 57.29 108. 0.58 109. 3.73 110. 1 111. 0.27 112. 5.67 113. 1.73 6 3 114. tan 40 = x 2 2 tan 40 = x x 1.68 ft 115. tan 60 = x 6 6 tan 60 = x x 10.39 ft 19
116. tan 20 = 15 x x = 15 tan 20 x 41.21 m x 117. tan 25 = 11 11 tan 25 = x 118. tan 63 = 3 2 x 119. tan X = 5 9 x 1.55 yd x = 3 2 tan 63 x 2.16 yd Ê m X = tan 1 5ˆ Ë Á 9 29.05 120. tan X = 43 30 Ê m X = tan 1 43ˆ Ë Á 30 55.10 121. tan X = 8 3 6 2 Ê ˆ 8 3 m X = tan 1 6 2 Ë Á 58.52 122. tan X = 49 15 Ê m X = tan 1 49ˆ Ë Á 15 72.98 123. tan X = 17.1 16.4 Ê m X = tan 1 17.1ˆ Ë Á 16.4 46.20 20
124. tan 23 = N 45 45 tan 23 = N N 19.10 mi 125. tan 35 = h 2450 2450 tan 35 = h 126. tan 70 = h 20 20 tan 70 = h h 1715.51 ft h 54.95 ft 127. tan x = 2.5 12 Ê x = tan 1 2.5ˆ Ë Á 12 11.77 The angle formed by the ramp and the ground is approximately 11.77. 128. tan x = 5.9 12.9 Ê x = tan 1 5.9 ˆ Ë Á 12.9 24.58 The angle formed by the ramp and the ground is approximately 24.58. 129. cosecant 130. inverse sine 131. sine 132. sin B = 3 3 = 6 133. sin C = 25 35 = 5 7 134. sin C = 2 2 15 3 2 135. sin D = 6 3 54 = 3 9 136. csc A = 12 8 = 3 2 137. csc A = 2 2 = 2 2 138. csc F = 25 15 = 5 3 21
139. csc F = 12 6 3 = 12 3 18 140. csc P = 50 16 = 25 8 141. 0.5 142. 0.71 143. 0.87 144. 0.26 145. 0.97 146. 0.09 147. 1 148. 1.15 149. 2 150. 3.86 151. 1.15 152. sin 40 = x 2 2sin 40 = x x 1.29ft = 2 3 3 153. sin 60 = x 6 6sin 60 = x x 5.20ft 154. sin20 = 15 x x = 15 sin20 x 43.86 m 155. sin25 = 11 x x = 11 sin25 x 7.85 yd 156. sin63 = 3 2 x x = 3 2 sin63 x 4.76 m 22
157. sin X = 8 15 Ê m X = sin 1 8 ˆ Ë Á 15 32.23 158. sin X = 30 42 Ê m X = sin 1 30ˆ Ë Á 42 45.58 159. sin X = 4 3 8 Ê ˆ 4 3 m X = sin 1 8 Ë Á 60 160. sin X = 1.1 5.2 Ê m X = sin 1 1.1ˆ Ë Á 5.2 12.21 161. sin18 = N 12 12sin18 = N N 3.71 mi 162. sin15 = E 23 23sin15 = E E 5.95 mi 163. sin35 = 12 l l = 12 sin 35 l 20.92ft 164. sin9 = 125 l l = 125 sin9 l 799.06ft 23
165. sinx = 45 100 Ê x = sin 1 45 ˆ Ë Á 100 26.74 The angle formed by the string and the ground is approximately 26.74. 166. sinx = 19 58 Ê x = sin 1 19ˆ Ë Á 58 19.12 The angle formed by the ramp and the ground is approximately 19.12. 167. sinx = 4 12 Ê x = sin 1 4 ˆ Ë Á 12 19.47 The angle formed by the bleachers and the ground is approximately 19.47. 168. sinx = 20 24 Ê x = sin 1 20ˆ Ë Á 24 56.44 The angle formed by the rope and the ground is approximately 56.44. 169. The inverse cosine of x is defined as the measure of an acute angle whose cosine is x. 170. cos B = 3 3 6 = 171. cos B = 7 14 = 1 2 172. cos C = 25 35 = 5 7 173. cos C = 2 2 15 174. cos D = 3 2 3 36 3 = 3 36 175. cos D = 6 3 54 = 3 9 176. sec A = 2 2 = 2 2 177. sec F = 25 20 = 5 4 24
178. sec F = 12 6 = 2 179. sec P = 3 5 6 PR 2 = 6 2 + 3 2 = 5 2 because PR 2 = 36 + 9 PR 2 = 45 PR = 45 = 3 5 180. sec P = 17 8 because 15 2 + PQ 2 = 17 2 225 + PQ 2 = 289 PQ 2 = 64 PQ = 8 181. 0.87 182. 0.71 183. 0.5 184. 0.97 185. 0.26 1 186. cos(45 ) = 1.41 187. 188. 189. 1 cos(25 ) = 1 1 cos(75 ) = 3.86 1 cos(60 ) = 2 190. cos 40 = x 2 2 cos 40 = x 191. cos 60 = x 6 6cos 60 = x x 1.53ft x = 3ft 25
192. cos 20 = 15 x x = 193. cos 5 = 2 x 15 cos 20 x 15.96 m x = 194. cos 25 = 2 cos 5 x 2.01 m x = 195. cos X = 9 13 11 x 11 cos 25 x 3.66 yd Ê m X = cos 1 9 ˆ Ë Á 13 46.19 196. cos X = 4 12 Ê m X = cos 1 4 ˆ Ë Á 12 70.53 197. XV 2 = 6 2 + 8 2 XV 2 = 36 + 64 XV 2 = 100 XV = 10 cos X = 6 10 Ê m X = cos 1 6 ˆ Ë Á 10 53.13 26
198. XD 2 + 3 2 = 8 2 XD 2 + 9 = 64 XD 2 = 55 XD = 55 cos X = 55 8 Ê ˆ m X = cos 1 55 8 Ë Á 22.02 199. XD 2 + 2 2 = 5 2 XD 2 + 4 = 25 XD 2 = 21 XD = 21 cos X = 21 5 Ê ˆ m X = cos 1 21 5 Ë Á 23.58 200. cos 21 = E 4230 4230 cos 21 = E 201. cos 76 = S 38 38cos 76 = S E 3949.05 ft S 9.19 km 202. cos 15 = 120 s s = 120 cos 15 s 124.23 ft 203. cos 45 = 15 l l = 15 cos 45 x 21.21 ft 27
204. cos 76 = x 24 x = 24 cos 76 x 5.81 ft The base of the ladder is approximately 5.81 feet from the edge of the house. 205. cos 34 = 9 x x cos 34 = 9 x = 9 cos 34 10.86 yd The length of the path is approximately 10.86 yd. 206. sin A = a c cos B = a c 207. tan D = d e cot E = d e 208. sec N = p m csc M = p m 209. tan S = s r cot R = s r 210. sec U = w v csc V = w v 211. I would use the sine ratio because the hypotenuse is given and the length of the side opposite the given angle needs to be determined. 212. I would use the cotangent ratio because the side opposite the given angle is given and the length of the side adjacent to the given angle needs to be determined. 213. I would use the secant ratio because the side adjacent to the given angle is given and the length of the hypotenuse needs to be determined. 214. I would use the tangent ratio because the side adjacent to the given angle is given and the length of the side opposite the given angle needs to be determined. 215. I would use the cosecant ratio because the side opposite the given angle is given and the length of the hypotenuse needs to be determined. 28
216. tan 15 = h 3 3 tan15 = h h 0.80 mi 217. tan 2 = 135 d d = 135 tan 2 d 3865.89ft 218. tan 1.5 = 151 d d = 151 tan 1.5 d 5766.46ft h 219. tan28 = 12,000 12,000 tan 28 = h h 6380.51ft 220. cos 67 = d 6 6 cos 67 = d d 2.34ft 221. cos 51 = 1.3 r r = 1.3 cos 51 r 2.07 m 222. tan34 = 25 d d = 25 tan 34 d 37.06ft 223. tan77 = 3500 d d = 3500 tan 77 d 808.04ft 29
224. tan 79 = d 3 3tan 79 = d d 15.43ft 225. tan 18 = h 250 250 tan 18 = h h 81.23ft 226. tan 1.6 = 50 d d = 50 tan 1.6 d 1790.03ft 227. tan 74 = 448 d d = 448 tan 74 d 128.46 m 228. sin 18 = 10 l l = 10 sin 18 l 32.36ft 229. sin 26 = 300 l l = 300 sin 26 l 684.35ft 230. The Law of Sines states that the ratios of the sines of each angle measure to the opposite sides are equal: sin A a = sin B b 231. A = 1 ab sin C 2 = sin C. c A = 1 (19)(16)(sin 67 ) 2 A 139.9 The area of the triangle is approximately 139.9 square centimeters. 30
232. A = 1 2 ac sinb A = 1 (5)(9)(sin 28 ) 2 A 10.6 The area of the triangle is approximately 10.6 square inches. 233. A = 1 df sin E 2 A = 1 (11.2)(6.5)(sin 85 ) 2 A 36.3 The area of the triangle is approximately 36.3 square centimeters. 234. A = 1 ef sin D 2 A = 1 (19.4)(15.2)(sin 71 ) 2 A 139.4 The area of the triangle is approximately 139.4 square millimeters. 235. A = 1 rs sint 2 A = 1 (45)(45)(sin 22 ) 2 A 379.3 The area of the triangle is approximately 379.3 square centimeters. 236. A = 1 xz sin Y 2 A = 1 2 (17)(10)(sin133 ) A 62.2 The area of the triangle is approximately 62.2 square inches. 31
237. sin A a = sin B b sin 50 x = sin 85 12 12 sin 50 = x sin 85 x = 12 sin 50 sin 85 238. sin A a x 9.2cm = sin C c sin 96 x = sin 28 8 8sin96 = x sin 28 x = 8 sin 96 sin 28 239. x 16.9 in. sin B b = sin C c sin 65 x = sin 33 9.5 9.5 sin 65 = x sin 33 x = 9.5 sin 65 sin 33 240. sin A a x 15.8 cm = sin C c sin 35 25.8 = sin 125 x x sin 35 = 25.8 sin 125 x = 25.8 sin 125 sin 35 x 36.8cm 32
241. m B = 180 72 45 = 63 sin A a = sin B b sin 72 x = sin 63 19 19 sin 72 = x sin 63 x = 19 sin 72 sin 63 242. sin B b x 20.3 in. = sin A a sin B 6 = sin 80 8 8 sin B = 6 sin 80 sin B = 6 sin 80 8 0.739 243. m B = sin 1 (0.739) 47.6 sin B b = sin C c sin B 11.6 = sin 28 9.4 9.4 sin B = 11.6 sin28 sin B = 11.6 sin 28 9.4 0.579 244. m B = sin 1 (0.579) 35.4 sin B b = sin A a sin B 19 = sin 57 23 23sin B = 19 sin 57 sin B = 19 sin 57 23 0.693 m B = sin 1 (0.693) 43.9 33
245. sin B b sin B 16 = sin C c = sin 110 25 25 sin B = 16 sin 110 sin B = 16 sin 110 25 0.601 246. m B = sin 1 (0.601) 36.9 sin B B = sin A a sin B 16.2 = sin 132 25.8 25.8 sin B = 16.2sin 132 sin B = 16.2 sin 132 25.8 0.467 m B = sin 1 (0.467) 27.8 247. c 2 = a 2 + b 2 2ab cos C c 2 = 14 2 + 17 2 2(14)(17) cos 82 c 2 = 196 + 289 476 cos 82 418.75 c = 418.75 c 20.5 cm 248. a 2 = b 2 + c 2 2bc cos A a 2 = 11.7 2 + 8.6 2 2(11.7)(8.67) cos 21 a 2 = 136.89 + 73.96 201.24 cos 21 22.98 a = 22.98 a 4.8 cm 249. c 2 = a 2 + b 2 2ab cos C a 2 = 16 2 + 12 2 2(16)(12) cos 130 a 2 = 256 + 144 384cos 130 646.83 a = 646.83 a 25.4 in. 34
250. b 2 = a 2 + c 2 2ac cos B b 2 = 21 2 + 8 2 2(21)(8) cos 145 b 2 = 441 + 64 336 cos 145 780.24 b = 780.24 b 27.9cm 35