Transcript Using Trigonometric Functions to find Angles

Word Problems:
Using
Trigonometric Functions
to find Angles
developed by Vicki Borlaug
Walters State Community College
Summer 2008
Example 1.) A scuba diver is on the top of the water and is going to
dive in a straight line to a treasure chest on the ocean floor. The floor
of the ocean is flat and the water is 70 feet deep. The treasure chest
sits 200 feet away from a point on the ocean floor that is directly below
the diver.
a.) Find the angle of depression the scuba diver must dive to swim
directly to the treasure chest.
b.) Suppose that first the diver swims towards the treasure chest on
the surface for 95 feet and then dives in a straight line to the treasure
chest. Find the angle of depression the diver must make to reach the
treasure chest with this approach.
Question developed by V. Borlaug
WSCC, 2008
Not drawn to scale.
Example 2.) A crane
has a 65 foot long arm.
The base of the arm is
100 feet high and the tip
of the arm is 137 feet
high.
Find the angle of
elevation the crane’s
arm.
Question developed by V. Borlaug
WSCC, 2008
Not drawn to scale.
Example 3.) The roof of the
Eiffel Tower is 301 meters high.
The antenna that sits on the
roof is 24 meters tall. A man is
standing on the ground 200
meters away from the base of
the tower. His eyes are 1.75
meters above the ground.
Location: Paris, France
Architect: Gustave Eiffel
Constructed: 1887-1889, designed to be
the entrance arch for the World’s Fair, the
Exposition Universelle
Some History: In 1944 during WW II when
the Allies were approaching France, Hitler
ordered General Dietrich von Choltitz to
demolish the tower. Von Choltitz
disobeyed the order.
a.) Find the angle of elevation
required for the man to look
directly at the roof of the Eiffel
Tower.
b.) Find the angle of elevation
required for the man to look
directly at the top of the
antenna on the Eiffel Tower.
Question developed by V. Borlaug
WSCC, 2008
http://en.wikipedia.org/wiki/Eiffel_Tower
Not drawn to scale.
Question developed by V. Borlaug
WSCC, 2008
Example 4.) A Navy fighter jet is lined up to land on an
aircraft carrier. The fighter jet is 6000 feet due east of
the aircraft carrier. The fighter jet is 2000 feet above sea
level and the flight deck of the carrier is 85 feet above
sea level.
Find the angle of depression the fighter jet must use to
approach the deck of the carrier in a straight line
approach.
Example 5.) A helicopter sits on the ground 850 feet due
east of a motion picture camera. Initially the camera is
pointed directly at the helicopter on the ground. The
helicopter will ascend vertically 700 feet, fly due west for
1050 feet, and then hover. The camera man will keep the
camera pointed at the helicopter during the flight.
Find the angle the
camera will need to
make in order to
follow the flight of
the helicopter to its
hovering position.
(You may assume the
camera lens is at ground
level.)
Not drawn to scale.
Question developed by V. Borlaug
WSCC, 2008
Example 6.) A five foot gate is in the middle of a
picket fence. Find the angle the gate must swing
open so that its outer edge is three (perpendicular)
feet from the original fence line.
Yard Sale
Yard Sale
Yard Sale
Example 7.) In a sailboat race the boats must travel east,
turn counterclockwise around a first buoy, and then head
towards a second buoy. The second buoy is 200 feet west
and 425 feet north of the first buoy.
A sailboat is headed due east towards the first buoy. How
many degrees must the sailboat change its heading in order
to go counterclockwise around the first buoy and head
toward the second buoy?
angle
θ
Not drawn to scale.
Question developed by V. Borlaug
WSCC, 2008
Example 8.) A tow truck has
a 12 foot arm that is attached
to the truck bed 4.5 feet
above ground level. A car
has a bumper 2.1 feet above
ground level. A chain with a
hook drops vertically 6.3 feet
from the end of the arm and
is attached to the bumper.
Find the arm’s angle of
elevation of the tow truck’s
arm.
Example 9: A boat is tied with a rope from its bow to an
edge of a dock. The rope is stretched tightly so that it
is forms in a straight line. The bow of the boat is 6.9
feet above the water line and the dock is 4.2 feet
above the water line. The bow of the boat is 3.8
horizontal feet from the edge of the dock.
a.) Draw a picture to represent this situation and label with
the drawing with the appropriate numbers.
b.) Find the angle of depression of the rope.
c.) Find the length of the rope.
(Use two decimal places in your answer and include the units.)
d.) Suppose a sailor pulls in the rope so that its length is
reduced by 1 ¾ feet. Assume that the rope still forms a
straight line. Draw a picture to represent this situation
and find the new angle of depression. (Think about what
numbers in your original drawing have changed and what numbers have stayed
the same.)
The end!
1)
A)
200
x°
95
70
70
70
tan x 
200
x  19.3
105
y°
B)
70
t an y 
105
y  33.7
2)
37
65
x°
100
37
sin x 
65
sin x  34.7
24
3)
x° y°
200
A)
301
tan x 
200
x  56.4
301
B)
325
tan y 
200
y  58.4
6000
4)
x°
1915
2000
85
1915
tan x 
6000
x  17.7
200
850
5)
700
700
x° c
200
H
850
700
tan x 
200
x  74.1
6)
x°
3
5
3
sin x 
5
x  36.9
7)
200
x°
425
x°
180-x
425
tan x 
200
180 - 64.8  115.2 
x  64.8
12
x° 3.9
8)
4.5
6.3
2.1
3.9
sin x 
12
x  19.0
9)
A)
(6.9 - 4.2)=
B)
x°
2.7 x°
3.8
C)
2.7  3.8  y
y  4.66 feet
2
2
2
2.7
tan x 
3.8
x  35.4
2.7
sin z 
4.66  1.75
z  68.1