Name: ________________________ Class: ___________________ Date: __________
ID: A
Ch 9 Questions
Use a trigonometric ratio to determine the value of x. Round your answer to the nearest tenth. 1.
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
Name: ________________________ 6. Use
ID: A
ABC to answer parts (a) and (b).
a.
Using trigonometric ratios, name the ratio
a in two different ways. b
b.
Using trigonometric ratios, name the ratio
c in two different ways. a
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
Name: ________________________ 11. Use
ID: A
ABC to answer parts (a) and (b).
a.
Using trigonometric ratios, name the ratio
b in two different ways. c
b.
Using trigonometric ratios, name the ratio
b in two different ways. a
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
Name: ________________________
ID: A
Mid-Chapter Test 13. In the figure,
ABE and
ACD are right triangles.
ABE and
ACD related? Justify your answer.
a.
How are
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.
e.
opposite-to-hypotenuse ratio for each triangle using A as the reference angle. Round adjacent-to-hypotenuse answers to the nearest thousandth. Calculate the
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 and 12
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
Name: ________________________
ID: A
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°.
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. 17.
18.
5
Name: ________________________
ID: A
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°
Use a trigonometric ratio to determine the value of x. Round your answer to the nearest tenth. 20.
21.
Use a trigonometric ratio to determine m∠A. Round your answer to the nearest tenth. 22.
6
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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 in two different ways. e
c.
Using trigonometric ratios, name the ratio
f in two different ways. d
d.
Using trigonometric ratios, name the ratio
e in two different ways. f
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
Name: ________________________
ID: A
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. b. c. d.
1 sec x 1 tanx (sin x)(cos x) 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. b. c. d.
110 feet 220 feet 1100 feet 2200 feet
9
Name: ________________________
ID: A
____ 34. What is the length of AB to the nearest tenth?
a. b. c. d.
1.8 meters 3.8 meters 4.0 meters 4.1 meters
____ 35. Which is closest to the value of the adjacent-to-hypotenuse ratio for a 45° angle?
a. b. c. d.
1.414 0.5 1 0.707
____ 36. What is m∠N to the nearest degree?
a. b. c. d.
16° 17° 63° 73°
10
Name: ________________________
ID: A
____ 37. What is the area of this triangular garden plot to the nearest tenth?
a. b. c. d.
428.5 square feet 225.0 square feet 214.0 square feet 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. b. c. d.
383.8 meters 443.2 meters 664.8 meters 1329.5 meters
____ 39. Which one of the following statements is true? a. b. c. d.
As the measure of an acute angle increases, the sine and cosine of the angle increase. As the measure of an acute angle increases, the sine and cosine of the angle decrease. As the measure of an acute angle increases, the sine and tangent of the angle increase. As the measure of an acute angle increases, the cosine and tangent of the angle decrease.
11
Name: ________________________
ID: A
____ 40. In the figure shown, cos P = 0.60. What is the length of PN ?
a. b. c. d.
0.25 centimeter 14.4 centimeters 40 centimeters 44 centimeters
____ 41. A proposed wheelchair ramp is shown. What is the rise of the ramp to the nearest inch?
a. b. c. d.
14 inches 12 inches 81 inches 181 inches
____ 42. Which ratio has the same value as sec ∠E ?
a. b. c. d.
cos ∠E cot ∠E cos ∠G csc ∠G
12
Name: ________________________
ID: A
____ 43. In the diagram shown, m∠B = 42° and AB = 25 feet. Which equation can be used to calculate the value of x?
a. b. c. d.
x = 25(sin 42°) x = 25(cos 42°) x = 25(tan 42°) sin 42° x= 25
____ 44. Which is NOT a valid conclusion you can draw about this figure?
a. b. c. d.
AEC ∼ BDC slope of AC = slope of BC AE BD = AC BC AE BD = AC DC
____ 45. Which could you use to calculate the length of RT ?
a. b. c. d.
Triangle Area Formula Pythagorean Theorem Law of Sines Law of Cosines
13
Name: ________________________
ID: A
____ 46. Which of the following can be used to determine m∠A?
a. b. c. d.
ÁÊ BC ˜ˆ˜ ˜˜ sin −1 ÁÁÁÁ ˜ Ë AC ¯ ÊÁ BC ˆ˜ ˜˜ cos −1 ÁÁÁÁ ˜˜ AC Ë ¯ Ê ˆ Á AC ˜˜ ˜˜ sin −1 ÁÁÁÁ ˜ Ë BC ¯ AC 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. b. c. d.
5.0 feet 5.6 feet 10.9 feet 25.7 feet
____ 48. What is the value of csc x, if sinx =
a. b. c. d.
5 ? 13
12 13 13 12 13 5 12 5
14
Name: ________________________
ID: A
____ 49. If cos A ≈ 0.67, which of the following statements must be true?
a. b. c. d.
The measure of ∠A is between 30° and 45°. The measure of ∠A is between 45° and 60°. The measure of ∠A is between 60° and 75°. The measure of ∠A is between 75° and 90°.
____ 50. Which of the following statements is NOT true? a. b. c. d.
The cosine of an acute angle is always less than or equal to one. The sine of an acute angle is always less than or equal to one. The tangent of an acute angle is always less than or equal to one. 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
Name: ________________________
ID: A
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
b.
Calculate the length of the hypotenuse of triangle ABC. Round your answer to the nearest tenth.
c.
opposite adjacent opposite , , and for the reference angle in triangle ABC. hypotenuse hypotenuse adjacent Round your answers to the nearest thousandth if necessary. Calculate the ratios
ABC and
d.
Describe the relationship between
e.
Calculate the length of the hypotenuse in your reasoning.
f.
g.
ABC .
DEF . Explain your reasoning.
DEF without using the Pythagorean Theorem. Explain
opposite adjacent opposite , , and for the reference angle in hypotenuse hypotenuse adjacent Round your answers to the nearest thousandth if necessary. Calculate the ratios
Compare the values of the three ratios for think this is true?
ABC and
DEF .
DEF . What do you observe? Why do you
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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.
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. 60.
61.
62.
18
Name: ________________________
ID: A
63. A bridge is shown in the following figure. Use the figure and the fact that 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.
AGC is congruent to
EGC to
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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 using ∠A as the reference angle in each triangle. Write your answers as hypotenuse fractions in simplest form.
Determine the ratio
70.
71.
21
Name: ________________________
ID: A
72.
73.
74.
adjacent using ∠A as the reference angle in each triangle. Write your answers as hypotenuse fractions in simplest form.
Determine the ratio
75.
22
Name: ________________________
ID: A
76.
77.
78.
79.
80.
23
Name: ________________________
ID: A
opposite adjacent opposite , , and using ∠A as the reference angle in each hypotenuse hypotenuse adjacent triangle. Write your answers as fractions in simplest form. Determine the ratios
81.
82.
83.
84.
24
Name: ________________________
ID: A
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 for reference angle A hypotenuse
86.
adjacent for reference angle A hypotenuse
87.
opposite for reference angle A hypotenuse
25
Name: ________________________
88.
adjacent for reference angle A hypotenuse
89.
opposite for reference angle A adjacent
90.
opposite for reference angle A adjacent
ID: A
26
Name: ________________________
ID: A
The Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent Vocabulary Match each description to its corresponding term for triangle EFG.
a. b. c.
tangent cotangent inverse tangent
____ 91.
EG in relation to ∠G EF
____ 92.
EF in relation to ∠G EG
ÊÁ EF ˆ˜ ˜˜ in relation to ∠G ____ 93. tan −1 ÁÁÁ ÁË EG ˜˜¯ Problem Set Calculate the tangent of the indicated angle in each triangle. Write your answers in simplest form. 94.
tan B = 95.
tan B =
27
Name: ________________________
ID: A
96.
tan C = 97.
tan C = 98.
tan D = 99.
tan D =
28
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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° Use a tangent ratio or a cotangent ratio to calculate the missing length of each triangle. Round your answers to the nearest hundredth. 114.
30
Name: ________________________
ID: A
115.
116.
117.
118.
Calculate the measure of angle X for each triangle. Round your answers to the nearest hundredth. 119.
31
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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. Problem Set Calculate the sine of the indicated angle in each triangle. Write your answers in simplest form. 132.
sin B = 133.
sin C =
34
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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° Use a sine ratio or a cosecant ratio to calculate the missing length of each triangle. Round your answers to the nearest hundredth. 152.
153.
154.
155.
37
Name: ________________________
ID: A
156.
Calculate the measure of angle X for each triangle. Round your answers to the nearest hundredth. 157.
158.
159.
160.
38
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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.
cos B = 171.
cos B = 172.
cos C =
41
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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
Name: ________________________
ID: A
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 Problem Set Determine the area of each triangle. Round your answers to the nearest tenth. 231.
232.
233.
50
Name: ________________________
ID: A
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
Name: ________________________
ID: A
240.
241.
Determine m∠B by using the Law of Sines. Round your answers to the nearest tenth. 242.
243.
52
Name: ________________________
ID: A
244.
245.
246.
Determine the unknown side length by using the Law of Cosines. Round your answers to the nearest tenth. 247.
248.
53
Name: ________________________
ID: A
249.
250.
54
ID: A Ch 9 Questions Answer Section 1. sin48° =
x 20
20(sin48°) = x 14.9 mm ≈ x 12 2. cos 39° = x x=
12 cos 39°
x ≈ 15.4in. x tan 64° = 15
3.
15(tan64°) = x 30.8ft ≈ x ÊÁ 5 ˆ˜ 4. m∠X = tan −1 ÁÁÁÁ ˜˜˜˜ ≈ 32° Ë 8¯ 5.
sin38° =
altitude 864
altitude = 864sin38° altitude ≈ 532 Katie' s kite is about 532 feet above the ground. sin40° =
altitude 725
altitude = 725sin40° altitude ≈ 466 Matt' s kite is about 466feet above the ground. a a tan∠A = andcot ∠B = 6. a. b b b.
sec ∠B =
c c andcsc ∠A = a a
1
ID: A
7. a.
A=
1 ab sinC 2
A=
1 (20)(13)(sin 110°) 2
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 The length of the longest side of the triangle is approximately 27.3 centimeters. x sin26° = 8
8.
8(sin26°) = x 3.5mm ≈ x 40 9. cos 58° = x x=
40 cos 58°
x ≈ 75.5 in. x tan71° = 27
10.
27(tan71°) = x 78.4 ft ≈ x 11. a.
sin ∠B =
b b and cos ∠A = c c
b.
tan ∠B =
b b and cot ∠A = a a
2
ID: A
12. a.
A=
1 ab sinC 2
A=
1 (10)(15)(sin 78°) 2
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
13. a.
b.
c.
d.
The length of the longest side of the triangle is approximately 16.2 inches. 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.
ABE , the ratio is
In
ACD, AD = 11.8cm + 9.5cm = 21.3 cm , so the ratio is
In
ABE , the ratio is
In
ACD, AC = 10.7cm + 8.6cm = 19.3 cm , so the ratio is
In In
e.
5.0 ≈ 0.424. 11.8
In
9.0 ≈ 0.423. 21.3
10.7 ≈ 0.907. 11.8
0.424 ≈ 0.467. 0.907 0.423 ≈ 0.467. ACD, the ratio is 0.906 ABE , the ratio is
The two ratios in each question are equal.
3
19.3 ≈ 0.906. 21.3
ID: A
14. a.
b.
c.
15. a.
b.
The ratio is
1 2.5 = . 37.5 15
1 1 < and 2.5 feet does not exceed 30 15 12 inches. Two and five-tenths feet is equal to 30 inches: 2.5× 12 = 30. Yes, Trevor’s ramp satisfies the requirements because
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. ÁÊ 2.5 ˜ˆ˜ ˜˜ m∠A = tan− 1 ÁÁÁÁ ˜ Ë 37.5 ¯
m∠A ≈ 3.8° 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. The height of the ski slope is approximately 1305.7 feet. x tan23° = 3076
3076(tan23°) = x 16. a.
1035.7 ≈ x The ladder reaches approximately 9.8 feet up the wall.
sin55° =
x 12
12(sin55°) = x 9.8 ≈ x 17. tan∠A =
28 13
ÊÁ 28 ˆ˜ m∠A = tan− 1 ÁÁÁ ˜˜˜˜ ≈ 65.1° ÁË 13 ¯ 25 18. cos ∠A = 45 ÁÊ 25 ˜ˆ m∠A = cos −1 ÁÁÁÁ ˜˜˜˜ ≈ 56.3° Ë 45 ¯
4
ID: A
19. a.
sin60° =
9 3 3 = 18 2
b.
csc 60° =
18 2 = 9 3 3
c.
sin30° =
9 1 = 18 2
d.
csc 30° =
18 =2 9
e.
sin45° =
10 1 = 10 2 2
f.
csc 45° =
10 2 = 10
20. sin55° =
x=
2
27 x 27 sin55°
x ≈ 33.0 ft 8 21. tan73° = x x=
8 tan73°
x ≈ 2.4 in. ÊÁ 5 ˆ˜ 22. m∠A = tan −1 ÁÁÁÁ ˜˜˜˜ ≈ 51.3° Ë 4¯ ÊÁ 10 ˆ˜ 23. m∠A = cos −1 ÁÁÁÁ ˜˜˜˜ ≈ 24.6° Ë 11 ¯ 24.
sin30° =
x 30
30(sin30°) = x 15 = x The height of the ramp is 15 feet. x cos 59° = 25. 37 37(cos 59°) = x 19.1 ≈ x The width of the rectangle is about 19.1 centimeters. 5
ID: A
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
ÊÁ 60 ˆ˜ m∠B = cos −1 ÁÁÁ ˜˜˜˜ ≈ 33.6° ÁË 72 ¯
27. a.
The measure of the angle formed where the ladder touches the top of the building is approximately 33.6°. Angles D and F are complementary.
b.
sin ∠D =
d d and cos ∠F = e e
c.
tan ∠F =
f f and cot ∠D = d d
d.
sec ∠D =
e e and csc ∠F = f f
28. A =
A=
1 ab (sinC) 2 1 (15)(15)(sin 28°) 2
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
ID: A
sinA sin B = b a
30.
sin75° sin B = 15 13 13sin75° = 15 sinB sinB =
13sin75° ≈ 0.837 15
m∠B ≈ 56.8° 31. a. I would use the Law of Cosines.
32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.
b.
I would use the Law of Sines.
c. D D B D B C B C C A D A D C A A C B C A
I would use the Law of Cosines.
7
ID: A
52. a.
The leg opposite ∠A is BC . The leg adjacent to ∠A is AC . The hypotenuse is AB.
b.
c2 = a2 + b2 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 7.0 = ≈ 0.574 hypotenuse 12.2 adjacent 10.0 = ≈ 0.820 hypotenuse 12.2 opposite 0.574 = = 0.7 adjacent 0.820
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 DE = AC AB 15.0 DE = 10.0 12.2 DE =
(15.0) (12.2) = 18.3 100
The length of the hypotenuse of f.
DEF is 18.3 centimeters.
opposite 10.5 = ≈ 0.574 hypotenuse 18.3 adjacent 15.0 = ≈ 0.820 hypotenuse 18.3 opposite 0.574 = = 0.7 adjacent 0.820 8
ID: A
g. 53.
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. x cot 72° = 12
12cot 72° = x x ≈ 3.9 in. ÊÁ 21 ˆ˜ 54. x = tan −1 ÁÁÁÁ ˜˜˜˜ Ë 23 ¯ x ≈ 42.4° 55.
csc 57° =
x 11
11csc 57° = x x ≈ 13.1ft ÁÊ 17 ˜ˆ 56. x = sin −1 ÁÁÁÁ ˜˜˜˜ Ë 25 ¯ x ≈ 42.8° ÊÁ 20 ˆ˜ 57. x = sin −1 ÁÁÁÁ ˜˜˜˜ Ë 58 ¯ x ≈ 20.2° 58.
sin50° =
x 75
75sin50° = x x ≈ 57.5m
9
ID: A
sin40° =
59. a.
CG 16
16sin40° = CG 10.28 ≈ CG The height of the roof truss is about 10.28 feet.
csc 40° =
b.
AB 6
6csc 40° = AB 9.33 ≈ AB The length of AB is about 9.33 feet. c.
ÁÊ 16 − 9.33 ˜ˆ˜ ˜˜ m∠BGC = sin −1 ÁÁÁÁ ˜ Ë 10.28 ¯ ÊÁ 6.67 ˆ˜ ˜˜ m∠BGC = sin −1 ÁÁÁÁ ˜˜ Ë 10.28 ¯ m∠BGC ≈ 40.45° The measure of angle BGC is about 40.45 degrees.
(CG) 2 = (BG) 2 + (BC) 2
d.
(10.28) 2 = (BG) 2 + (6.67) 2 (BG) 2 = 61.1895 BG ≈ 7.82
60.
The length of the support beam is about 7.82 feet. x cos 33° = 15
15cos 33° = x x ≈ 12.6m ÊÁ 12 ˆ˜ 61. x = cos −1 ÁÁÁÁ ˜˜˜˜ Ë 19 ¯ x ≈ 50.8° ÁÊ 17 ˜ˆ 62. x = cos −1 ÁÁÁÁ ˜˜˜˜ Ë 22 ¯ x ≈ 39.4°
10
ID: A
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
23.5 sec 27° = CH 26.4 ≈ CH The length of CH is about 26.4 feet. d.
ÊÁ 12 ˆ˜ ˜˜ m∠BHC = cos −1 ÁÁÁ ÁË 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
ID: A 64. a.
b.
Minimum
10,000 tan35° = x
Maximum 10,000 tan33° = x
x tan35° = 10,000
x tan33° = 10,000
x=
10,000 tan35°
x=
x ≈ 14,281.5
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 10,000 sin35° = y
y sin35° = 10,000 y=
Maximum 10,000 sin33° = y
y sin33° = 10,000
10,000 sin35°
y=
y ≈ 17,434.5
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
ID: A 65. a.
b.
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. for Emily’s design: 1 A = ac sinB 2
=
1 (11)(8)(sin 80°) 2
≈ 43.3 The area for Emily’s design is approximately 43.3 square feet. for Joe’s design: 1 A = ac sinB 2
=
1 (11)(8)(sin110°) 2
≈ 41.3 The area for Joe’s design is approximately 41.3 square feet. c.
for Emily’s design: b 2 = a 2 + c 2 − 2ac cos B
for Emily’s design: b 2 = a 2 + c 2 − 2ac cos B
b 2 = 11 2 + 8 2 − 2(11)(8) cos 80°
b 2 = 11 2 + 8 2 − 2(11)(8) cos 110°
b 2 = 121 + 64 − 176cos 80° ≈ 154.4
b 2 = 121 + 64 − 176cos 110° ≈ 245.2
b=
b=
154.4
b ≈ 12.4
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
ID: A 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.
m∠B = 180° − 85° − 42° = 53°
b.
sinA sinB = b a sin 85° sin63° = a 11 11sin85° = a sin63° a=
11sin85° sin53°
a ≈ 13.7cm A= =
1 ab sinC 2 1 (13.7)(11)(sin 42°) 2
≈ 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 = 13.72 + 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
67. 68. 69. 70. 71.
For Chris’s design, the perimeter of the play space is approximately 33.9 feet. reference angle adjacent side opposite side opposite 6 3 = = hypotenuse 10 5
opposite 24 12 = = hypotenuse 26 13
14
ID: A 72. c 2 = a 2 + b 2
c 2 = 15 2 + 82 c 2 = 225 + 64 = 289 c = 289 = 17 opposite 15 = hypotenuse 17 73. c 2 = a 2 + b 2
c 2 = 7 2 + 242 c 2 = 49 + 576 = 625 c = 625 = 25 opposite 7 = hypotenuse 25 74. c 2 = a 2 + b 2
Ê c 2 = 1 2 + ÁÁÁ Ë
ˆ2 3 ˜˜˜ ¯
c2 = 1 + 3 = 4 c= 4 =2 opposite 1 = hypotenuse 2 75.
adjacent 20 4 = = hypotenuse 25 5
76.
adjacent 16 8 = = hypotenuse 34 17
77.
c2 = a2 + b2 c 2 = 1.4 2 + 4.8 2 c 2 = 1.96 + 23.04 = 25.0 c=
25.0 = 5.0
adjacent 4.8 24 = = hypotenuse 5.0 25
15
ID: A
c2 = a2 + b2
78.
c 2 = 42 + 42 c 2 = 16 + 16 = 32 c=
32 = 4 2
adjacent 2 4 1 = = or 2 hypotenuse 4 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 1.0 5 = = hypotenuse 2.6 13 80. c 2 = a 2 + b 2
c 2 = 2 2 + (2 3) 2 c 2 = 4 + 12 = 16 c=
16 = 4
adjacent 2 3 3 = = hypotenuse 4 2 81.
opposite 18 3 = = hypotenuse 30 5 adjacent 24 4 = = hypotenuse 30 5 opposite 18 3 = = adjacent 24 4
16
ID: A 82. b 2 = c 2 − a 2
b 2 = 51 2 − 24 2 b 2 = 2601 − 576 = 2025 b=
2025 = 45
opposite 24 8 = = hypotenuse 51 17 adjacent 45 15 = = hypotenuse 51 17 opposite 24 8 = = adjacent 45 15 83. a 2 = c 2 − b 2
a 2 = 29 2 − 20 2 a 2 = 841 − 400 = 441 a=
441 = 21
opposite 21 = hypotenuse 29 adjacent 20 = hypotenuse 29 opposite 21 = adjacent 20 84. a 2 = c 2 − b 2
b 2 = (5 2) 2 − 5 2 b 2 = 50 − 25 = 25 b=
25 = 5
opposite 5 2 5 = or hypotenuse 2 2 adjacent 5 2 5 = or hypotenuse 2 2 opposite 5 = =1 adjacent 5
17
ID: A 85.
AE = 4 + 4 = 8 AD = 5 + 5 = 10 opposite 3 In ABC, = . hypotenuse 5 opposite 6 3 In ADE, = = . hypotenuse 10 5
86.
AE = 15 + 30 = 45 AD = 17 + 34 = 51 adjacent 15 In ABC, = . hypotenuse 17 adjacent 45 15 In ADE, = = . hypotenuse 51 17
87.
AE = 10 + 15 = 25 AD = 10 2 + 15 2 = 25 2
88.
In
ABC,
opposite 2 10 1 = = or . 2 hypotenuse 10 2 2
In
ADE,
opposite 2 25 1 = = or . 2 hypotenuse 25 2 2
AE = 2 3 +
3 =3 3
AD = 4 + 2 = 6
In
ADE,
adjacent 2 3 3 = = . hypotenuse 4 2
In
ADE,
adjacent 3 3 3 = = . 6 2 hypotenuse
89. In
ABC,
opposite 8 = . adjacent 15
In
ADE,
opposite 24 8 = = . adjacent 45 15
90. In
ABC,
opposite 2.1 21 = = . adjacent 2.0 20
In
ADE,
opposite 8.4 21 = = . adjacent 8.0 20
91. B 92. A 93. C 94. tan B =
2 =1 2
18
ID: A
95. tan B =
3 2
98. tan D =
2 2 15
99. tan D =
3
=1 3 2 25 5 = 96. tan C = 20 4 32 4 = 97. tan C = 40 5
5 5 4 100. cot A = 3 6 3 101. cot A = = 8 4 7 102. cot F = 15
=
3 5 25
2 6 6 = 6 3 32 4 cot A = = 40 5 1.73 0.27 57.29 0.58 3.73 1 0.27 5.67 1.73 x tan 40° = 2
103. cot F = 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114.
2 tan 40° = x
115.
x ≈ 1.68 ft x tan 60° = 6 6 tan 60° = x x ≈ 10.39 ft
19
ID: A
116. tan 20° =
x=
117.
15 x 15 tan 20°
x ≈ 41.21 m x tan 25° = 11 11 tan 25° = x x ≈ 1.55 yd
118. tan 63° =
x=
3 2 x 3 2 tan 63°
x ≈ 2.16 yd 5 119. tan X = 9 ÁÊ 5 ˜ˆ m∠X = tan −1 ÁÁÁÁ ˜˜˜˜ ≈ 29.05° Ë9¯ 120. tan X =
43 30
ÊÁ 43 ˆ˜ m∠X = tan −1 ÁÁÁ ˜˜˜˜ ≈ 55.10° ÁË 30 ¯ 121. tan X =
8 3 6 2
ÊÁ ˆ ÁÁ 8 3 ˜˜˜ Á ˜˜ ≈ 58.52° m∠X = tan ÁÁ ÁÁ 6 2 ˜˜˜ Ë ¯ 49 122. tan X = 15 ÁÊ 49 ˜ˆ m∠X = tan −1 ÁÁÁÁ ˜˜˜˜ ≈ 72.98° Ë 15 ¯ −1
123. tan X =
17.1 16.4
ÊÁ 17.1 ˆ˜ ˜˜ ≈ 46.20° m∠X = tan −1 ÁÁÁ ÁË 16.4 ˜˜¯
20
ID: A
124.
tan 23° =
N 45
45 tan 23° = N
125.
N ≈ 19.10 mi h tan 35° = 2450 2450 tan 35° = h
126.
h ≈ 1715.51 ft h tan 70° = 20 20 tan 70° = h
h ≈ 54.95 ft 2.5 127. tan x = 12 ÊÁ 2.5 ˆ˜ ˜˜ ≈ 11.77° x = tan −1 ÁÁÁÁ ˜˜ 12 Ë ¯ The angle formed by the ramp and the ground is approximately 11.77°. 5.9 128. tan x = 12.9 ÊÁ 5.9 ˆ˜ ˜˜ ≈ 24.58° x = tan −1 ÁÁÁÁ ˜˜ Ë 12.9 ¯ The angle formed by the ramp and the ground is approximately 24.58°. 129. cosecant 130. inverse sine 131. sine
3 3 3 = 6 2 25 5 = 133. sin C = 35 7 132. sin B =
134. sin C =
2 2 15
6 3 3 = 54 9 12 3 = 136. csc A = 8 2 135. sin D =
2 2 = 2 25 5 = 138. csc F = 15 3 137. csc A =
2
21
ID: A
12 3 2 3 12 = = 18 3 6 3 50 25 csc P = = 16 8 0.5 0.71 0.87 0.26 0.97 0.09 1 1.15 2 3.86 1.15 x sin 40° = 2
139. csc F = 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152.
2sin 40° = x
153.
x ≈ 1.29ft x sin 60° = 6 6sin 60° = x
x ≈ 5.20ft 15 154. sin20° = x x=
15 sin20°
x ≈ 43.86 m 155. sin25° =
x=
11 x 11 sin25°
x ≈ 7.85 yd 156. sin63° =
x=
3 2 x 3 2 sin63°
x ≈ 4.76 m
22
ID: A
157. sin X =
8 15
ÁÊ 8 ˜ˆ m∠X = sin −1 ÁÁÁÁ ˜˜˜˜ ≈ 32.23° Ë 15 ¯ 30 158. sin X = 42 ÊÁ 30 ˆ˜ m∠X = sin −1 ÁÁÁÁ ˜˜˜˜ ≈ 45.58° Ë 42 ¯ 4 3 8 ÊÁ ˆ ÁÁ 4 3 ˜˜˜ ˜˜ ≈ 60° m∠X = sin −1 ÁÁÁ ÁÁ 8 ˜˜˜ Ë ¯ 1.1 160. sin X = 5.2 ÊÁ 1.1 ˆ˜ ˜˜ ≈ 12.21° m∠X = sin −1 ÁÁÁÁ ˜˜ 5.2 Ë ¯ 159. sin X =
161.
sin18° =
N 12
12sin18° = N
162.
N ≈ 3.71 mi E sin15° = 23 23sin15° = E
E ≈ 5.95 mi 12 163. sin35° = l l=
12 sin 35°
l ≈ 20.92ft 125 164. sin9° = l l=
125 sin9°
l ≈ 799.06ft
23
ID: A
165. sinx =
45 100
ÁÊ 45 ˜ˆ˜ ˜˜ ≈ 26.74° x = sin −1 ÁÁÁÁ ˜ Ë 100 ¯ The angle formed by the string and the ground is approximately 26.74°. 19 166. sinx = 58 ÁÊ 19 ˜ˆ x = sin −1 ÁÁÁÁ ˜˜˜˜ ≈ 19.12° Ë 58 ¯ The angle formed by the ramp and the ground is approximately 19.12°. 4 167. sinx = 12 ÁÊ 4 ˜ˆ x = sin −1 ÁÁÁÁ ˜˜˜˜ ≈ 19.47° Ë 12 ¯ The angle formed by the bleachers and the ground is approximately 19.47°. 20 168. sinx = 24 ÊÁ 20 ˆ˜ x = sin −1 ÁÁÁÁ ˜˜˜˜ ≈ 56.44° Ë 24 ¯ 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.
3 3 6 7 = 171. cos B = 14 25 = 172. cos C = 35 170. cos B =
173. cos C =
2 2 15
174. cos D =
3
175. cos D =
1 2 5 7
36 3
=
3 36
6 3 3 = 54 9
2 2 = 2 25 5 = 177. sec F = 20 4 176. sec A =
3 2
=
2
24
ID: A
178. sec F =
12 =2 6
3 5 5 = because 6 2 PR 2 = 6 2 + 3 2
179. sec P =
PR 2 = 36 + 9 PR 2 = 45 45 = 3 5 17 180. sec P = because 8 15 2 + PQ 2 = 172 PR =
225 + PQ 2 = 289 PQ 2 = 64 PQ = 8 181. 182. 183. 184. 185.
0.87 0.71 0.5 0.97 0.26 1 186. = 1.41 cos(45°) 187.
1 =1 cos(25°)
188.
1 = 3.86 cos(75°)
189.
1 =2 cos(60°)
190.
cos 40° =
x 2
2 cos 40° = x
191.
x ≈ 1.53ft x cos 60° = 6 6cos 60° = x x = 3ft
25
ID: A
192. cos 20° =
x=
15 x 15 cos 20°
x ≈ 15.96 m 2 193. cos 5° = x x=
2 cos 5°
x ≈ 2.01 m
11 x
194. cos 25° =
x=
11 cos 25°
x ≈ 3.66 yd 9 195. cos X = 13 ÊÁ 9 ˆ˜ m∠X = cos −1 ÁÁÁÁ ˜˜˜˜ ≈ 46.19° Ë 13 ¯ 196. cos X =
4 12
ÊÁ 4 ˆ˜ m∠X = cos −1 ÁÁÁÁ ˜˜˜˜ ≈ 70.53° Ë 12 ¯ 197. XV 2 = 6 2 + 8 2 XV 2 = 36 + 64 XV 2 = 100 XV = 10 cos X =
6 10
ÊÁ 6 ˆ˜ m∠X = cos −1 ÁÁÁ ˜˜˜˜ ≈ 53.13° ÁË 10 ¯
26
ID: A 198. XD 2 + 3 2 = 8 2
XD 2 + 9 = 64 XD 2 = 55 XD =
55
cos X =
55 8
ÁÊÁ Á m∠X = cos −1 ÁÁÁ ÁÁ Ë 2 2 2 199. XD + 2 = 5
55 8
˜ˆ˜ ˜˜ ˜˜ ≈ 22.02° ˜˜ ¯
XD 2 + 4 = 25 XD 2 = 21 XD =
21
cos X =
21 5
ÊÁ ˆ ÁÁ 21 ˜˜˜ Á ˜˜ ≈ 23.58° m∠X = cos ÁÁ ÁÁ 5 ˜˜˜ Ë ¯ E cos 21° = 4230 −1
200.
4230 cos 21° = E
201.
E ≈ 3949.05 ft S cos 76° = 38 38cos 76° = S
S ≈ 9.19 km 120 202. cos 15° = s s=
120 cos 15°
s ≈ 124.23 ft 15 203. cos 45° = l l=
15 cos 45°
x ≈ 21.21 ft
27
ID: A
x 24
204. cos 76° =
x = 24 cos 76° x ≈ 5.81 ft The base of the ladder is approximately 5.81 feet from the edge of the house. 9 cos 34° = 205. x
x cos 34° = 9 x=
9 ≈ 10.86 yd cos 34°
The length of the path is approximately 10.86 yd. a 206. sin ∠A = c
a c d 207. tan ∠D = e cos ∠B =
d e p 208. sec ∠N = m cot ∠E =
csc ∠M = 209. tan ∠S =
p m
s r
s r w 210. sec ∠U = v cot ∠R =
w v 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. 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. 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. 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. 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. csc ∠V =
211. 212. 213. 214. 215.
28
ID: A
216.
tan 15° =
h 3
3 tan 15° = h h ≈ 0.80 mi 135 217. tan 2° = d d=
135 tan 2°
d ≈ 3865.89ft 151 218. tan 1.5° = d d=
219.
151 tan 1.5°
d ≈ 5766.46ft h tan28° = 12,000
12,000 tan 28° = h
220.
h ≈ 6380.51ft d cos 67° = 6 6 cos 67° = d
d ≈ 2.34 ft 1.3 221. cos 51° = r r=
1.3 cos 51°
r ≈ 2.07 m 25 222. tan34° = d d=
25 tan 34°
d ≈ 37.06ft 3500 223. tan77° = d d=
3500 tan 77°
d ≈ 808.04 ft
29
ID: A
224.
tan 79° =
d 3
3tan 79° = d d ≈ 15.43 ft h tan 18° = 250
225.
250 tan 18° = h h ≈ 81.23ft 50 226. tan 1.6° = d d=
50 tan 1.6°
d ≈ 1790.03ft 448 227. tan 74° = d 448 tan 74°
d=
d ≈ 128.46 m 10 228. sin 18° = l l=
10 sin 18°
l ≈ 32.36 ft 300 229. sin 26° = l l=
300 sin 26°
l ≈ 684.35 ft 230. The Law of Sines states that the ratios of the sines of each angle measure to the opposite sides are equal: sin A sin B sin C = = . a b c 1 231. A = ab sin C 2 A=
1 (19)(16)(sin 67°) 2
A ≈ 139.9 The area of the triangle is approximately 139.9 square centimeters.
30
ID: A
232. A =
A=
1 ac sinB 2 1 (5)(9)(sin 28°) 2
A ≈ 10.6 The area of the triangle is approximately 10.6 square inches. 1 233. A = 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. 1 234. A = 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. 1 235. A = rs sinT 2
A=
1 (45)(45)(sin 22°) 2
A ≈ 379.3 The area of the triangle is approximately 379.3 square centimeters. 1 236. A = xz sin Y 2
A=
1 (17)(10)(sin 133°) 2
A ≈ 62.2 The area of the triangle is approximately 62.2 square inches.
31
ID: A
237.
sin A sin B = b a sin 50° sin 85° = x 12 12 sin 50° = x sin 85° x=
238.
12 sin 50° sin 85°
x ≈ 9.2cm sin A sin C = a c sin 96° sin 28° = 8 x 8sin 96° = x sin 28° x=
239.
8 sin 96° sin 28°
x ≈ 16.9 in. sin B sin C = b c sin 65° sin 33° = x 9.5 9.5 sin 65° = x sin 33° x=
240.
9.5 sin 65° sin 33°
x ≈ 15.8 cm sin A sin C = a c sin 35° sin 125° = 25.8 x x sin 35° = 25.8 sin 125° x=
25.8 sin 125° sin 35°
x ≈ 36.8cm
32
ID: A 241.
m∠B = 180° − 72° − 45° = 63° sin A sin B = a b sin 72° sin 63° = 19 x 19 sin 72° = x sin 63° x=
242.
19 sin 72° sin 63°
x ≈ 20.3 in. sin B sin A = b a sin B sin 80° = 6 8° 8 sin B = 6 sin 80° sin B =
243.
6 sin 80° ≈ 0.739 8
m∠B = sin −1 (0.739) ≈ 47.6° sin B sin C = b c sin B sin 28° = 11.6 9.4° 9.4 sin B = 11.6 sin28° sin B =
244.
11.6 sin 28° ≈ 0.579 9.4
m∠B = sin −1 (0.579) ≈ 35.4° sin B sin A = b a sin B sin 57° = 19 23° 23sin B = 19 sin 57° sin B =
19 sin 57° ≈ 0.693 23
m∠B = sin −1 (0.693) ≈ 43.9°
33
ID: A
245.
sin B sin C = c b sin B sin 110° = 16 25° 25 sin B = 16 sin 110° sin B =
246.
16 sin 110° ≈ 0.601 25
m∠B = sin −1 (0.601) ≈ 36.9° sin B sin A = a B sin B sin 132° = 16.2 25.8° 25.8 sin B = 16.2sin 132° sin B =
16.2 sin 132° ≈ 0.467 25.8
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 − 384 cos 130° ≈ 646.83 a=
646.83
a ≈ 25.4 in.
34
ID: A 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
