4.3 Right Triangle Trigonometry Day 2 Objective: Use right triangle trigonometry to solve real life problems.

Warm-up Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Quiz Wed. on sections 4.1-4.4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Degrees, Minutes, and Seconds There is another way to state the size of an angle, one that subdivides a degree into smaller pieces. In a full circle there are 360 degrees. Each degree can be divided into 60 parts, each part being 1/60 of a degree. These parts are called minutes. Each minute can divided into 60 parts, each part being 1/60 of a minute. These parts are called seconds. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Degrees, Minutes, and Seconds Conversions To convert decimal degrees into DMS, multiply decimal degrees by 60 To convert from DMS to decimal degrees, divide minutes by 60, seconds by 3600 OR use the Angle feature of your calculator

Examples

Vocabulary Angle of Depression and Angle of Elevation are Equal Horizontal Line Angle of Depression and Angle of Elevation are Equal

Applying Trig Functions You sight a rock climber on a cliff at a 32o angle of elevation. The horizontal ground distance to the cliff is 1000 ft. Find the line of sight distance to the rock climber. x 1000 ft

Example: An airplane pilot sights a life raft at a 26o angle of depression. The airplane’s altitude is 3 km. What is the airplane’s horizontal distance d from the raft? 3 km d

Applying Trig Functions A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as How tall is the tree? Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Applying Trig Functions You are 200 yards from a river. Rather than walking directly to the river, you walk 400 yards, diagonally, along a straight path to the rivers edge. Find the acute angle between this path and the river’s edge. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example: A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is 20 feet from the streetlight and 8 feet from the tip of the streetlight’s shadow, the person’s shadow starts to appear beyond the streetlight’s shadow. What is the height of the streetlight? Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Applying Trig Functions You are standing 90 feet away from a building. You estimate that the angle of elevation to the top of the building is 80 degrees. What is the approximate height of the building? One of your friends is at the top of the building sightseeing. What is the distance between you and your friend?

Question developed by V. Borlaug WSCC, 2008 Example: A helicopter is hovering 560 feet above a straight and level road that runs east and west. On the road east of the helicopter are a car and a truck. From the helicopter the angle of depression to the car is 48º and the angle of depression to the truck is 38º. Find the distance between the car and the truck. Question developed by V. Borlaug WSCC, 2008 Not drawn to scale.

H 560 T C X Y Tan 38 = 560/y Tan 48 = 560/x Y = 716.8 X = 504.2 48° Answers H 38° 560 48 38 T X C Y Tan 48 = 560/x X = 504.2 Tan 38 = 560/y Y = 716.8 Distance = 716.8 – 504.2 = 212.6

Example: A helicopter takes off from ground level and goes 853 feet with an angle of elevation of 23º. The helicopter then changes course and goes 719 feet with and angle of elevation of 49º and then it hovers in this position. Find the height of the hovering helicopter.

Answers 719 Y 49° 853 X 23° Sin 23 = X/853 X = 333.3 Sin 49 = Y/719 Y = 542.6 333.3 + 542.6 = 875.9

Example: A short building is 200 feet away from a taller building Example: A short building is 200 feet away from a taller building. Jessie is on the roof of the short building. To see the top of the taller building requires Jessie to look up with a 38 angle of elevation. To see the bottom of the taller building requires Jessie to look down with 12 angle of depression. Find the height of the taller building. (You may assume that Jessie’s height is negligible.) (not dawn to scale)

Answers 200 X 38° 12° Y Tan 38 = X/200 X = 156.3 Tan 12 = Y/200 Y = 42.5 156.3 + 42.5 = 198.8

Example: In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Homework Right Triangle Applications Worksheet

Finding a Side of a Right Triangle (given a side and an angle) Word Problems: Finding a Side of a Right Triangle (given a side and an angle) developed by Vicki Borlaug Walters State Community College Summer 2008 Note to Instructor: These word problems do not require Law of Sines or Law of Cosines.

Question developed by V. Borlaug WSCC, 2008 Not drawn to scale. Question developed by V. Borlaug WSCC, 2008 Example 8.) A boat is due south of a light house. From his charts the captain knows that the light house is 0.76 miles east of the pier. The captain takes measurements from his boat and finds an angle of 55º from the light house to the pier. a.) Find the distance from the boat to the pier. b.) Find the distance from the boat to the light house.

8) Sin 55 = .76/x X = .93 Tan 55 = .76/y Y = .53 .76 P L Y X 55° B Answers 8) Sin 55 = .76/x X = .93 .76 P L Y X 55° B Tan 55 = .76/y Y = .53

Question developed by V. Borlaug WSCC, 2008 Not drawn to scale. Example 1.) A straight boat ramp is being designed to cover a horizontal distance of 150 feet with an angle of elevation of 3º. Find the length of the boat ramp.

Question developed by V. Borlaug WSCC, 2008 Example 2.) A hot air balloon rises vertically. Katie is standing on the level ground 20 feet from a point on the ground where the balloon was launched. She points her camera at the balloon and takes a picture. When Katie takes the picture the camera is 5.5 feet off the ground and has a 80º angle of elevation . Find the height of the hot air balloon when the picture is taken. Question developed by V. Borlaug WSCC, 2008 Not drawn to scale.

X = 150.2 1) Cos 3 ° = 150/x X 3° 2) Tan 80 ° = X/20 X = 113.4 Answers 1) Cos 3 ° = 150/x X = 150.2 X 3° 150 2) Tan 80 ° = X/20 X = 113.4 113.4 + 5.5 = 118.9 X 80° 20

Question developed by V. Borlaug WSCC, 2008 Example 4.) Derek is scuba diving in still water. Starting from the surface he dives in a straight line with a 65º angle of depression. Derek is traveling at a constant rate of 25 feet per minute along this straight line path. a.) Find the distance Derek has traveled along this straight line path three minutes into the dive. b.) Find Derek’s vertical depth three minutes into the dive. c.) Find the rate at which Derek’s vertical depth is changing as he descends. Question developed by V. Borlaug WSCC, 2008

X 75 c) 68.0 / 3 = 22.7 ft per min 4) a) 3 x 25 = 75 b) Sin 65 = X Answers 65° X 75 c) 68.0 / 3 = 22.7 ft per min 4) a) 3 x 25 = 75 b) Sin 65 = X X = 68.0 75

Question developed by V. Borlaug WSCC, 2008 Example 5.) A river runs between a tree and a lamp post. The tree is directly north of the lamp post. The surveyor is 324 feet east of the lamp post. He measures an angle of 11.3º from the tree to the lamp post. Find the distance from the surveyor to the tree. Question developed by V. Borlaug WSCC, 2008 Not drawn to scale.

Answers T X 11.3° L S 324 5) Cos 11.3 = 324 X X = 330.4

Question developed by V. Borlaug WSCC, 2008 Example 3.) A crane has a 100 foot arm with a hook at the end of the arm. This crane is designed so that the angle of elevation of the arm can be changed. When the crane’s hook is attached to an object on the ground, the arm’s angle of elevation is 5º. The crane’s arm then rotates upward and raises the object to a position that gives the arm a 25º angle of elevation. Find the height the crane has lifted the object. HINT: Find the tip of the arm’s initial distance from horizontal. Then find the distance from horizontal after the object has been lifted. Use these to find the height the object has been lifted. Not drawn to scale. Question developed by V. Borlaug WSCC, 2008

100 X 5° 3) Sin 5 ° = X/100 X = 8.7 Sin 25 = Y/100 Y = 42.3 Answers 3) Sin 5 ° = X/100 X = 8.7 Sin 25 = Y/100 Y = 42.3 Height = 42.3 – 8.7 Height = 33.6 Feet 100 X 5° 100 Y 25°

Question developed by V. Borlaug WSCC, 2008 Example 6.) Jessica is standing 75 feet from the base of a vertical tree. She is 5 ½ feet tall and her eyes are 4 inches from the top of her head. It takes a 38º angle of elevation for Jessica to look at the top of the tree. (Use four decimal place accuracy.) a.) Find the height of the tree in feet. b.) Find the height of the tree in meters. NOTE: One meter is the length equal to 1,650,763.73 wavelengths in a vacuum of the orange-red radiation of krypton 86. One meter also equals 39.37 inches. Ref: The American Heritage Dictionary of the English Language, American Publishing Co., Inc., 1969 Question developed by V. Borlaug WSCC, 2008 Not drawn to scale.

6) X 38° 75 5 ft 2 inches Cos 38 = 75/x X = 95.1764 5’ 2” = 5.1667ft Answers X 38° 75 5 ft 2 inches 6) Cos 38 = 75/x X = 95.1764 5’ 2” = 5.1667ft Height = 95.1764 + 5.1667 = 100.3431 feet or 100.3431 / 39.37 = 2.5488 meters

Question developed by V. Borlaug WSCC, 2008 Example 9.) A straight 67 inch ramp is being designed for a skate board course. a.) Find the vertical height required to give the skate board ramp a 75º angle of depression. b.) Find the vertical height required to give the skate board ramp a 65º angle of depression. c.) Find the vertical height required to give the skate board ramp a 55º angle of depression. Question developed by V. Borlaug WSCC, 2008

Sin 65 = X/67 X = 60.7 Sin 75 = X Sin 55 = X/67 67 X = 64.7 X = 54.9 X Answers Sin 65 = X/67 X = 60.7 67 X 75° 9) Sin 75 = X 67 X = 64.7 Sin 55 = X/67 X = 54.9

Example 12: Solar panels are being installed on a roof Example 12: Solar panels are being installed on a roof. The roof has an angle of elevation of 9.2º. Each solar panel is 7.20 feet long and 4.80 feet wide. The longer edge of each panel will be horizontally installed on the roof. The shorter edge of the panel will be installed with an angle of elevation 29.6º. a.) Find the vertical height of each solar panel relative to horizontal. b.) Find the vertical height of each solar panel relative to the roof directly the solar panel’s highest edge. Hint for part “b”: Using the two angles of elevation there here are two right triangles. First solve the larger right triangle completely.

Answers 4.80 X cos 29.6 = Y/4.8 Y = 4.17 Z 29.6° 9.2° Y 12) Sin 29.6 = X 4.8 X = 2.4 tan 9.2 = Z/4.17 Y = 0.7 2.4 – 0.7 = 1.7