rtlWVOU'1..1.. I..EA~N • To use trigonometry to

solve problems involving angles of elevation or depression.

rtlJYIf'S IMPOltfANf You can use angles of elevation and depression to find missing measures involved in aerospace, architecture, and meteorology.

Example

Architecture

Angles of Elevation and Depression

Tourism

Suppose Arnoldo is on the Skydeck of the Sears Tower looking through a telescope at Hanna who is on the Observation Deck of the John Hancock Center. Hanna is looking through a telescope at Arnoldo. An angle is formed by a horizontal line between the John Hancock Center and the Sears Tower and the line of sight from Hanna to Arnoldo. This angle is called the angle of elevation. Another angle is formed by a horizontal line between the Sears Tower and the John Hancock Center and the line of sight from Arnoldo to Hanna. This angle is called the angle of depression.

angle of depression

Arnalda

angle of elevation

Sometimes drawing a diagram of the situation described in a problem can help you to solve problems involving angles of elevation and depression.

o The Observation Deck of the John Hancock Center is on the 94th floor, which is 1030 feet above the ground. The Skydeck of the Sears Tower is on the 103rd floor, which is 1335 feet from the ground. The Jqhn Hancock Center is 1.7 miles or 8976 feet from the Sears Tower. What is the angle of elevation from Hanna to Arnoldo?

Explore The problem gives the height of the Observation Deck and the height of the Skydeck. It also gives the distance between the John Hancock Center and the Sears Tower. It asks for the angle of elevation from Hanna to Arnoldo.

Plan Draw a diagram. 1335 It A (Arnoldo)

~1335-1030=3051t H(Hanna) R 1030 It ~1.7 mi or 8976 It---I

The length of one leg of the right triangle is 1. 7 miles or 8976 feet. The length of the other leg is 1335 - 1030 or 305 feet. The angle of elevation can be found by using tan -1 H.

420 Chapter 8 Applying Right Triangles and Trigonometry

tan 25°

n tan 25°

ftan 18°

305So{v ' tan H = 8976

ENTER: 305 [3 8976 ~ I 2nd! [tan-I] 1= I /.'3'-16/332

The angle of elevation is about 1.9°.

76Emrnine Since tan A = 8;05 , use a calculator to find mLA.

ENTER: 8976 ~ 305 r=J _2nd' [tan-I] C=j 88.053867

Since 1.9461332 + 88.053867 = 90, the angles are complementary, and the answer is verified .

...............................................................................................................................................................................

Angles of elevation or depression to two different objects can be used to find the distance between those objects.

Example On July 20, 1969, Neil Armstrong became the first human to walk on the moon. During this mission, the lunar lander Eagle traveled aboard Apollo 11. Before sending Eagle to the surface of the moon, Apollo 11 orbited the moon three miles above the surface. At one point in the orbit, the onboard guidance system measured the angles of depression

Aerospace to the far and near edges of a large crater. The angles measured 18° and 25°, respectively. Find the distance across the crater.

orbit

25'

----f n~1

Let fbe the ground distance from Apollo 11 to the far edge of the crater and n be the ground distance to the near edge. ~

3 Iun 0pjJosrtetan 18° = f udjuC/!/I1

= 3 Cross multiply

f=_3_ Diuisiorl Propel1y (=)tan 18'

3 [3 18 ITANJ LJ 9.2330506 Hak(' \111'(' YOllr calculafor IS

III deuree mod . 3 (tl!.P'Isi/(J

= ­n tan - odjn,,>nl

= 3 Cross multiply. 3 n=-­ DIL'i.~1011 Prop 'rly ( -)tan 25'

3 [3 251TANI 5.'-1335208

Since f = 9.2 and n = 6.4, the distance across the crater is about 9.2 - 6.4 or 2.8 miles .

...............................................................................................................................................................................

Lesson 8-4 Angles of Elevation and Depression 421

~HECK FOR UNDERSTANDI.NG. __ .

Communicating Mathematics

Study the lesson. Then complete the following.

1. Describe in your own words the meaning of angle of depression.

2. Draw an example showing an angle of elevation. Identify the angle of elevation.

3. Explain how you decide whether to use sin, cos, or tan when you are finding the measure of an acute angle in a right triangle.

~RNAL 4. Assess Yourself Name three common trigonometric ratios and describe each ratio. What is the meaning of each ratio? When you solve a problem involving trigonometric ratios, do you find a diagram helpful? What suggestions can you make to help your classmates solve these types of problems?

Guided 5. Name the angles of elevation ;rpPractice and depression in the figure at ----------------~O

the right.

State an equation that would enable you to solve each problem. Then solve. Round answers to the nearest tenth.

6. Given mLP = 15 and PO = 37, find OR. Q

7. Given PR = 2.3 and PO = 5.5, find mLP. ~ P R

Refer to the chart at the right for Exercises 8-9.

8. Charo is 50 feet from the tallest totem pole. If Charo's eyes are 5 feet from the ground, find the angle of elevation for her line of sight to the top of the totem.

9. Derrick is visiting the San Jacinto State Park outside Houston, Texas. The angle of elevation for hisThe Son Jacinto Column is the

tallest monumental (olumn in line of sight to the top of the the world. San Jacinto Column is 75°. If

his eyes are 6 feet from the ground, how far is he from the base of the column?

Monument Heights San Jacinto Column 570 feet

n'ear Houston

Gateway to the West Arch 630 feet St. Louis

1 I Washington Monument 555 feet

Washington, D,C,

Statu of Liberty 305 feet New York City \(

J' ­

Tallest totem pole J73 feet Alberta Bay, Canada

Source: Com,oarisons

10. Aviation The cloud ceiling is the lowest altitude at which solid cloud is present. If the cloud ceiling is below a certain level, usually about 61 meters, airplanes are not allowed to take off or land. One way that meteorologists can find the cloud ceiling at night is to shine a searchlight straight up and observe the spot of light on the clouds from a location away from the searchlight.

a. If the searchlight is located 200 meters from the meteorologist and the angle of elevation to the spot of light on the clouds is 35°, how high is the cloud ceiling?

b. Can the airplanes land and take off under these conditions? Explain.

422 Chapter 8 Applying Right Triang'les and Trigonometry

11. H .---------------~~~

12.~:,

13.

b?1H

B F

JI

K

14. T

u

E~XERCISES

Practice Name the angles of elevation and depression in each figure.

F

State an equation that would enable you to solve each problem. Then solve. Round answers to the nearest tenth.

15. Given YZ = 28 andXZ = 54, find mLY. y

16. Given XY = 15 and mLX = 28, find yz.

17. Given mLY = 66 and YZ = 7, find XY.

18. Given YZ = 4 and XY = 15, find mLY.

19. Given XZ = 4.5 and XY = 6.6, find mLX.

20. Given XY = 22.4 and mL Y = 65.5, find Xz. X

21. The tallest fountain in the world is located at Fountain Hills, Arizona. If weather conditions are favorable, the water column can reach 625 feet. Suppose Alfonso visits the fountain on a perfect day and his eyes are 5 feet from the ground. c. If Alfonso stands 40 feet from the fountain, find the angle of elevation for

his line of sight to the top of the spray. b. If Alfonso moves so that the angle of elevation for his line of sight to the

top of the spray is 75°, how far is he from the base of the spray?

22. After flying at an altitude of 9 kilometers, an airplane starts to descend when its ground distance from the landing field is 175 kilometers. What is the angle of depression for this portion of the flight? ~

23. A golfer is standing on a tee with the green in a valley below. If the tee is 43 yards higher than the green and the angle of depression from the tee to the hole is 14°, find the distance from the green to the hole.

24. Kierra is flying a kite. She has let out 55 feet of string. If the string makes a 35° angle with the ground, how high above the ground is the kite?

25. A trolley car track rises vertically 40 feet over a horizontal distance of 630 feet. What is the angle of elevation of the track?

26. A ski slope is 550 yards long with a vertical drop of 130 yards. Find the angle of depression of the slope.

Critical Thinking

Applications and Problem Solving

27. The waterway between Lake Huron and Lake Superior separates the United States and Canada at Sault Sainte Marie. The railroad drawbridge located at Sault Saint Marie is normally 13 feet above the water when it is closed. Each section of this drawbridge is 210 feet long. Suppose the angle of elevation of each section is 70°.

a. Find the distance from the top of a section of the drawbridge to the water.

b. Find the width of the gap created by the two sections of the bridge.

28. Carol is in the Skydeck of the Sears Tower overlooking Lake Michigan. She sights two sailboats going due east from the tower. The angles of depression to the two boats are 42° and 29°. If the Skydeck is 1335 feet high, how far apart are the boats?

29. Ulura or Ayers Rock is a sacred place for Aborigines of the western desert of Australia. Chun-Wei uses a surveying device to measure the angle of elevation to the top of the rock to be 11.so. He walks half a mile closer and measures the angle of elevation to be 23.9°. How high is Ayers Rock in feet?

30. Imagine that a fly and an ant are in one corner I

of a rectangular box. The end of the box is IFood I

4 inches by 6 inches, and the diagonal across r---~...------~ Insects ------".",/ ..... "'" the bottom of the box makes an angle of 21.8° 4in. // ......... ­

with the longer edge of the box. There is food :::>- 21.8'

in the corner opposite the insects. 6 in. a. What is the shortest distance the fly must

fly to get to the food? b. What is the shortest distance the ant must

crawl to get to the food?

31. Given acute 6GME with altitude EH, E

write a trigonometric expression for . GE

the ratIO EM' &G H M

32. Literature In The Adventures ofSherlock Holmes: The Adventures of the Musgrave Ritual, Sherlock Holmes uses trigonometry to solve the mystery. To find a treasure, he must determine where the end of the shadow of an elm tree was located at a certain time of day. Unfortunately, the elm had been cut down, but Mr. Musgrave remembers that his tutor required him to calculate the height of the tree as part of his trigonometry class. Mr. Musgrave tells Sherlock Holmes that the tree was exactly 64 feet. Sherlock needs to find the length of the shadow at a time of day when the shadow from an oak tree is a certain length. The angle of elevation of the sun at this time of day is 33.r. What was the length of the shadow of the elm?

424 Chapter 8 Applying Righi Triangles and Trigonometry

33. Architecture Diana is an architect who designs houses so that the windows receive minimum sun in the summer and maximum sun in the winter. For Seattle, Washington, the angle of elevation of the sun at noon on the longest day is 66° and on the shortest day is 19°. Suppose a house is designed with a south-facing window that is 6 feet tall. The top of the window is to be installed 1 foot below the overhang. a. How long should Diana make the overhang

so that the window gets no direct sunlight at noon on the longest day?

b. Using the overhang from part a, how'much of the window will get direct sunlight at noon on the shortest day?

c. To find the angle of elevation of the sun on the longest day of the year where you live, subtract your latitude from 90° and add 23.5°. To find the elevation of the sun on the shortest day, subtract the latitude from 90° and then subtract 23.5°. Draw a solar design for a south-facing window and corresponding overhang for a home in your community.

34. Meteorology Two weather observation stations are 7 miles apart. A weather balloon is located between the stations. From Station 1, the angle of elevation to the weather balloon is 35°. From Station 2, the angle of elevation to the balloon is 54°. Find the altitude of the balloon to the nearest tenth of a mile. (Hint: Find the distance from Station 2 to the point directly below the balloon.)

35. Find the indicated trigonometric ratio as a fraction and as a decimal rounded to the nearest ten-thousandth. (Lesson 8 -3) 2:::J'ma. sin A b. cos B c. tan A

B 16cm C

36. The perimeter of an equilateral triangle is 42 centimeters. Find the length of an altitude of the triangle. (Lesson 8-2)

37. Construction Find the length of a diagonal brace needed for a rectangular section of wall that is 6 feet wide and 8 feet high. (Lesson 8 -1)

38. Photography Chapa wants to enlarge a photograph that is currently 4 inches wide by 5 inches long so tp.at the new photograph is 12 inches long. How wide will the new photograph be? (Lesson 7-2)

39. Draw a trapezoid with two right angles and one obtuse angle. (Lesson 6-5)

40. Quadrilateral l¥XYZ is a parallelogram. If WX = 3g + 7, XY = 7h - 1, YZ = 6g - 2, and WZ = 2h + 9, find the perimeter of WXYZ. (Lesson 6-1)

41. The base of an isosceles triangle is 18 inches long. If the legs are 3y + 21 and lOy inches long, find the perimeter of the triangle. (Lesson 4-6)

( 42. Find the slope and y-intercept of the graph of 2x - y = 16.

Algebra 43. Solve the system of inequalities by graphing. y<5 y>2x+1

Lesson 8-4 Angles of Eleva/ion and Depression 425