Download Angles of Elevation and Depression

Angles of Elevation and
Depression
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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.
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
angle of elevation
Arnalda
Sometimes drawing a diagram of the situation described in a problem can
help you to solve problems involving angles of elevation and depression.
Example
Architecture
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 I t - - - 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
So {v '
tan H =
305
8976
305
ENTER:
[3
8976 ~
I 2nd!
[tan-I]
1= I
/.'3'-16/332
The angle of elevation is about 1.9°.
Emrnine
Since tan A
76
= 8;05 ,
use a calculator to find mLA.
8976 ~ 305
ENTER:
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
Aerospace
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
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.
~
tan 18°
=
3
Iu n
f
udjuC/!/I1
ftan 18° = 3
Cross multiply
f=_3_
tan 18'
3
[3
0pjJosrte
Diuisiorl Propel1y (= )
I
18 TANJ
LJ
9.2330506
Hak(' \111'(' YOllr calculafor IS
deuree mod .
III
3
(tl!.P'Isi/(J
n
tan - odjn,,>nl
= 3
3
n=­
Cross multiply.
tan 25°
= ­
n tan 25°
tan 25'
3
[3
DIL'i.~1011
251TANI
Prop 'rly ( -)
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
Guided
Practice
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?
5. Name the angles of elevation
and depression in the figure at
the right.
;rp
----------------~O
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.
The Son Jacinto Column is the
tallest monumental (olumn in
the world.
9. Derrick is visiting the San Jacinto
State Park outside Houston, Texas.
The angle of elevation for his
line of sight to the top of the
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
630 feet
Gateway to the West Arch
St. Louis
1
Washington Monument
Washington, D,C,
Statu of Liberty
New York City
Tallest totem pole
Alberta Bay, Canada
555 feet
305 feet
\(
J'­
J73 feet
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
I
E~XERCISES
Practice
Name the angles of elevation and depression in each figure.
11.
H
.---------------~~~
F
12.~:,
B
13.
F
14.
b?1
H
T
JI
K
u
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 m LX = 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.
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?
Critical
Thinking
30. Imagine that a fly and an ant are in one corner
I
I
of a rectangular box. The end of the box is
Food
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,
write a trigonometric expression for
. GE
t h e ratIO
EM'
E
&
G
Applications and
Problem Solving
424
H
M
32. Literature In The Adventures of Sherlock 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?
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)
a. sin A
b. cos B
c. tan A
B
2:::J'm
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
Algebra
=
16.
43. Solve the system of inequalities by graphing.
y<5
y>2x+1
Lesson 8-4
Angles of Eleva/ion and Depression
425