Three Angle Measure. Introduction to Trigonometry. LESSON 9.1 Skills Practice. Vocabulary. Problem Set. Use the diagram to complete each sentence.
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1 LESSON.1 Skills Practice Name Date Three ngle Measure Introduction to Trigonometry Vocabulary Use the diagram to complete each sentence. 1. If b is the opposite side, then is the. y c 2. If y is the reference angle, then b is the. b 3. If is the reference angle, then b is the. Problem Set opposite Determine the ratio using / as the reference angle in each triangle. Write your answers as hypotenuse fractions in simplest form. a opposite hypotenuse hapter Skills Practice 671
2 LESSON.1 Skills Practice page adjacent Determine the ratio using / as the reference angle in each triangle. Write your answers as hypotenuse fractions in simplest form adjacent hypotenuse hapter Skills Practice
3 LESSON.1 Skills Practice page 3 Name Date hapter Skills Practice 673
4 LESSON.1 Skills Practice page 4 opposite Determine the ratios hypotenuse, adjacent hypotenuse, and opposite using / as the reference angle in each adjacent triangle. Write your answers as fractions in simplest form opposite hypotenuse adjacent hypotenuse opposite adjacent hapter Skills Practice
5 LESSON.1 Skills Practice page 5 Name Date In each figure, triangles and DEF are similar by the Similarity Theorem. alculate the indicated ratio twice, first using n and then using nde. 1. opposite hypotenuse for reference angle 20. adjacent hypotenuse E D opposite In n, 3 hypotenuse 5 5. opposite In nde, hypotenuse D E for reference angle 34 D E 24 hapter Skills Practice 675
6 LESSON.1 Skills Practice page opposite for reference angle hypotenuse 25 D adjacent 22. for reference angle hypotenuse D E E opposite for reference angle adjacent opposite 24. for reference angle adjacent D D E E 676 hapter Skills Practice
7 LESSON.2 Skills Practice Name Date The Tangent Ratio Tangent Ratio, otangent Ratio, and Inverse Tangent Vocabulary Match each description to its corresponding term for triangle EFG. F E G 1. EG in relation to /G EF a. tangent 2. EF in relation to /G EG b. cotangent 3. tan 21 ( EF EG ) in relation to /G c. inverse tangent hapter Skills Practice 677
8 LESSON.2 Skills Practice page 2 Problem Set alculate the tangent of the indicated angle in each triangle. Write your answers in simplest form ft 2 ft ft 3 2 ft tan tan m 40 m 20 m 32 m tan 5 tan m D 6. 3 ft 2 2 m 5 5 ft tan D 5 tan D 5 D 678 hapter Skills Practice
9 LESSON.2 Skills Practice page 3 Name Date alculate the cotangent of the indicated angle in each triangle. Write your answers in simplest form ft cot ft 8. 6 ft cot 5 8 ft. F yd 6 yd 15 yd cot F 5 cot F yd F ft m 4 2 ft 40 m cot 5 cot 5 Use a calculator to approimate each tangent ratio. Round your answers to the nearest hundredth. 13. tan tan tan tan tan tan 8 hapter Skills Practice 67
10 LESSON.2 Skills Practice page 4 Use a calculator to approimate each cotangent ratio. Round your answers to the nearest hundredth. 1. cot cot cot cot cot 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 ft ft tan tan 40 5 < 1.68 ft m m 680 hapter Skills Practice
11 LESSON.2 Skills Practice page 5 Name Date yd yd 25 alculate the measure of angle X for each triangle. Round your answers to the nearest hundredth X in. 5 in. 30 m X 5 tan X 5 m/x 5 tan 21 ( 5 ) < 2.05 T S 43 m R 33. K 8 3 cm M 34. Y 6 2 cm 5.4 km X Z 5.66 km X hapter Skills Practice 681
12 LESSON.2 Skills Practice page X 36. E 15 in yd 17.1 yd U 4 in. V X F Solve each problem. Round your answers to the nearest hundredth. 37. boat travels in the following path. How far north did it travel? N tan 23 5 N miles 45 tan 23 5 N N < 1.10 mi 38. During a group hike, a park ranger makes the following path. How far west did they travel? 2 miles 12 N 3. surveyor makes the following diagram of a hill. What is the height of the hill? ft 682 hapter Skills Practice
13 LESSON.2 Skills Practice page 7 Name Date 40. To calculate the height of a tree, a botanist makes the following diagram. What is the height of the tree? ft 41. moving truck is equipped with a ramp that etends from the back of the truck to the ground. When the ramp is fully etended, it touches the ground 12 feet from the back of the truck. The height of the ramp is 2.5 feet. alculate the measure of the angle formed by the ramp and the ground. 2.5 ft 12 ft? hapter Skills Practice 683
14 LESSON.2 Skills Practice page park has a skateboard ramp with a length of 14.2 feet and a length along the ground of 12. feet. The height is 5. feet. alculate the measure of the angle formed by the ramp and the ground.? 14.2 ft 12. ft 5. ft 43. lifeguard is sitting on an observation chair at a pool. The lifeguard s eye level is 6.2 feet from the ground. The chair is 15.4 feet from a swimmer. alculate the measure of the angle formed when the lifeguard looks down at the swimmer.? 6.2 ft 15.4 ft 44. surveyor is looking up at the top of a building that is 140 meters tall. His eye level is 1.4 meters above the ground, and he is standing 10 meters from the building. alculate the measure of the angle from his eyes to the top of the building. 140 m? 10 m 1.4 m 684 hapter Skills Practice
15 LESSON.3 Skills Practice Name Date The Sine Ratio Sine Ratio, osecant Ratio, and Inverse Sine Vocabulary Write the term from the bo that best completes each statement. sine cosecant inverse sine 1. 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. 2. The of is the measure of an acute angle whose sine is. 3. 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 alculate the sine of the indicated angle in each triangle. Write your answers in simplest form ft 3 3 ft 6 ft 14 ft sin sin m 25 m 15 m 2 2 m sin 5 sin 5 hapter Skills Practice 685
16 LESSON.3 Skills Practice page 2 5. D m 3 m 36 3 m 54 m D sin D 5 sin D 5 alculate the cosecant of the indicated angle in each triangle. Write your answers in simplest form ft 8 ft 2 2 ft csc ft csc 5. F yd 20 yd 6 3 yd 12 yd 15 yd 6 yd F 11. csc F 5 csc F 5 P mm 3 3 m 50 mm P 4 2 m csc P 5 csc P hapter Skills Practice
17 LESSON.3 Skills Practice page 3 Name Date Use a calculator to approimate each sine ratio. Round your answers to the nearest hundredth. 13. sin sin sin sin sin sin 5 Use a calculator to approimate each cosecant ratio. Round your answers to the nearest hundredth. 1. csc csc csc csc csc 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 ft 6 ft 60 sin sin 40 5 < 1.2 ft hapter Skills Practice 687
18 LESSON.3 Skills Practice page m 5 2 m yd m alculate the measure of angle X for each triangle. Round your answers to the nearest hundredth ft X H 32. X R 8 ft 42 mm R 30 mm sin X m/x 5 sin 21 ( 8 15 ) < T 688 hapter Skills Practice
19 LESSON.3 Skills Practice page 5 Name Date 33. L 34. X E 5 in. 17 in. 4 3 yd 8 yd F K X 35. X 36. X M 1.1 cm 25 ft 5.2 cm N D 20 ft Solve each problem. Round your answers to the nearest hundredth. 37. scout troop traveled 12 miles from camp, as shown on the map below. How far north did they travel? N sin 18 5 N miles 12 sin 18 5 N N < 3.71 mi hapter Skills Practice 68
20 LESSON.3 Skills Practice page n ornithologist tracked a ooper s hawk that traveled 23 miles. How far east did the bird travel? 23 miles N n 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? ft 40. 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? 125 ft 60 hapter Skills Practice
21 LESSON.3 Skills Practice page 7 Name Date 41. 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. alculate the measure of the angle formed by the string and the ground. 45 ft 100 ft? 42. n airplane ramp is 58 feet long and reaches the cockpit entrance 1 feet above the ground, as shown in the diagram. alculate the measure of the angle formed by the ramp and the ground. 1 ft 58 ft? hapter Skills Practice 61
22 LESSON.3 Skills Practice page leachers in a stadium are 4 meters tall and have a length of 12 meters, as shown in the diagram. alculate the measure of the angle formed by the bleachers and the ground. 4 m 12 m? foot flagpole is raised by a 24-foot rope, as shown in the diagram. alculate the measure of the angle formed by the rope and the ground. 24 ft 20 ft? 62 hapter Skills Practice
23 LESSON.4 Skills Practice Name Date The osine Ratio osine Ratio, Secant Ratio, and Inverse osine Vocabulary Describe the similarities and differences between the pair of terms. 1. cosine ratio and secant ratio Define the term in your own words. 2. inverse cosine Problem Set alculate the cosine of the indicated angle in each triangle. Write your answers in simplest form ft ft 6 ft cos ft cos m 35 m 15 m 2 2 m cos 5 cos 5 hapter Skills Practice 63
24 LESSON.4 Skills Practice page D 6 3 m 36 3 m 3 m 54 m D cos D 5 cos D 5 alculate the secant of the indicated angle in each triangle. Write your answers in simplest form ft ft 2 ft 8 ft sec sec 5. F yd 25 yd 6 3 yd 12 yd 15 yd sec F 5 sec F 5 6 yd F P 3 R 12. P Q sec P 5 sec P 5 64 hapter Skills Practice
25 LESSON.4 Skills Practice page 3 Name Date Use a calculator to approimate each cosine ratio. Round your answers to the nearest hundredth. 13. cos cos cos cos cos cos 8 Use a calculator to approimate each secant ratio. Round your answers to the nearest hundredth. 1. sec sec 25 1 cos(45 ) sec sec sec sec 60 Use a cosine ratio or a secant ratio to calculate the missing length of each triangle. Round your answers to the nearest hundredth ft 40 cos cos ft 60 < 1.53 ft hapter Skills Practice 65
26 LESSON.4 Skills Practice page m m yd yd alculate the measure of angle X for each triangle. Round your answers to the nearest hundredth. 31. D in. X V 10 m V in. X cos X m m/x 5 cos 21 ( ) < D 66 hapter Skills Practice
27 LESSON.4 Skills Practice page 5 Name Date 33. V D 34. V 12 ft 4 ft 8 cm X D 6 cm X 35. D 3 mm V X 8 mm 36. X 5 yd D V 2 yd hapter Skills Practice 67
28 LESSON.4 Skills Practice page 6 Solve each problem. Round your answers to the nearest hundredth. 37. The path of a model rocket is shown below. How far east did the rocket travel? N cos 21 5 E ft 4230 cos 21 5 E 21 E < ft 38. n ichthyologist tags a shark and charts its path. Eamine his chart below. How far south did the shark travel? N 38 km kite is flying 120 feet away from the base of its string, as shown below. How much string is let out? ft 40. pole has a rope tied to its top and to a stake 15 feet from the base. What is the length of the rope? ft 68 hapter Skills Practice
29 LESSON.4 Skills Practice page 7 Name Date 41. You park your boat at the end of a 20-foot dock. You tie the boat to the opposite end of the dock with a 35-foot rope. The boat drifts downstream until the rope is etended as far as it will go, as shown in the diagram. What is the angle formed by the rope and the dock? 35 ft downstream 20 ft? Dock 42. Rennie is walking her dog. The dog s leash is 12 feet long and is attached to the dog 10 feet horizontally from Rennie s hand, as shown in the diagram. What is the angle formed by the leash and the horizontal at the dog's collar? Leash 12 ft? 10 ft hapter Skills Practice 6
30 LESSON.4 Skills Practice page 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? House Ladder 24 ft 76? 44. rectangular garden 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. Garden? path 34 yd 700 hapter Skills Practice
31 LESSON.5 Skills Practice Name Date We omplement Each Other! omplement ngle Relationships Problem Set For each right triangle, name the given ratio in two different ways F d E c a e f b D a c a sin / 5 c a cos / 5 c d e 3. M n p 4. R s T P m N t r S p m s r 5. X y Z m Y 6. V u w W v U y z w v hapter Skills Practice 701
32 LESSON.5 Skills Practice page 2 Determine the trigonometric ratio that you would use to solve for in each triangle. Eplain your reasoning. You do not need to solve for cm in. 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 m 17 yd mm ft 702 hapter Skills Practice
33 LESSON.5 Skills Practice page 3 Name Date Solve each problem. Round your answers to the nearest hundredth. 13. You are standing 40 feet away from a building. The angle of elevation from the ground to the top of the building is 57. What is the height of the building? tan 57 5 h tan 57 5 h h 61.5 ft 14. 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? 15. The angle of elevation from a ship to a 135-foot-tall lighthouse is 2. How far is the ship from the lighthouse? 16. 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? hapter Skills Practice 703
34 LESSON.5 Skills Practice page The angle of elevation from the top of a person s shadow on the ground to the top of the person is 45. The top of the shadow is 50 inches away from the person. How tall is the person? 18. 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? 1. 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? 20. 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? 704 hapter Skills Practice
35 LESSON.5 Skills Practice page 5 Name Date Solve each problem. Round your answers to the nearest hundredth. 21. 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? tan d 25 d 5 tan 34 d ft 22. n 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? 23. 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 7. What is the depth of the well water? 24. 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? hapter Skills Practice 705
36 LESSON.5 Skills Practice page 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? 26. The Empire State uilding is 448 meters tall. The angle of depression from the top of the Empire State uilding to the base of the UN building is 74. How far is the UN building from the Empire State uilding? 27. factory conveyor has an angle of depression of 18 and drops 10 feet. How long is the conveyor? 28. 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? 706 hapter Skills Practice
37 LESSON.6 Skills Practice Name Date Time to Derive! Deriving the Triangle rea Formula, the Law of Sines, and the Law of osines Vocabulary Define each term in your own words. 1. Law of Sines 2. Law of osines Problem Set Determine the area of each triangle. Round your answers to the nearest tenth cm ab sin cm (1)(16)(sin 67 ) in. 5 in. The area of the triangle is approimately 13. square centimeters. hapter Skills Practice 707
38 LESSON.6 Skills Practice page 2 3. E 4. F 1.4 mm D 71 D 6.5 cm 85 F 11.2 cm E 15.2 mm 5. R S 45 cm 45 cm 22 T 6. Y 17 in. 10 in. 133 X Z Determine the unknown side length by using the Law of Sines. Round your answers to the nearest tenth cm sin a 5 sin b sin 50 5 sin sin 50 5 sin sin 50 sin 85.2 cm 8 in hapter Skills Practice
39 LESSON.6 Skills Practice page 3 Name Date cm cm in. hapter Skills Practice 70
40 LESSON.6 Skills Practice page 4 Determine m/ by using the Law of Sines. Round your answers to the nearest tenth cm 8 in. 14 cm in. sin b 5 sin a sin 5 sin sin 5 6 sin 80 sin 5 6 sin m/ 5 sin 21 (0.73) cm in..4 cm 23 in. 710 hapter Skills Practice
41 LESSON.6 Skills Practice page 5 Name Date in cm cm 16 in. Determine the unknown side length by using the Law of osines. Round your answers to the nearest tenth in. 5 in cm cm b 2 5 a 2 1 c 2 2 2ac cos b (5)(7)cos 42 b cos b b 4.7 in. hapter Skills Practice 711
42 LESSON.6 Skills Practice page cm 6.7 cm 11.7 cm 8.6 cm cm 16 in. 8 cm in hapter Skills Practice
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