Transcript 8.7 Applications of Right Triangle Trigonometry

8.4 Angles of Elevation and
Depression
-Quiz Friday over Pythagorean Theorem/Special
Right Triangles/Trig Ratios.
• Theodolite/diopter
• Inclinometer / Clinometer
Sextant – used to measure the angle between
two known objects, usually using a celestial
object such as the sun or Polaris (north star).
The major use is to identify lines of latitude
and sighting the height of a landmark to
determine distance off from the landmark
 Angle of Elevation  if the object being observed is above the horizontal then the angle
between the line of sight and the horizontal is called the Angle of Elevation.
Homemade Inclinometer. To read you simply align the flat part of the
protractor with the top of the landmark, and then the string will always be
perpendicular to the ground, the string will then pass through your
protractor this will then provide you with an angle that you can use in your
trig equations to find the height of an object. Record the angle (0-90) and
subtract that from 90, and that will give you the angle of elevation.
Some critical terminology
 Horizontal  any line constructed so that it is parallel with the
horizon or another horizontal line.
 Line of Sight  the line from the observer’s eye to the object
 Angle of Elevation  if the object being observed is above the
horizontal then the angle between the line of sight and the
horizontal is called the Angle of Elevation.
 Angle of Depression  if the object being observed is below the
horizontal then the angle between the line of sight and the
horizontal is called the Angle of Depression.
 Angle of Inclination  if the line of sight follows a physical object,
such as an inclined plane or a hillside, we use the term Angle of
Inclination.
Diagram of terminology.
Angle of Depression
Horizontal
Horizontal
Angle of Elevation
What is a description of the angle as it relates to the situation shown?
Angle 1?
Angle 3?
Angle 2?
Angle 4?
bird
person
Base of mountain
The hardest part of these story problems is drawing the picture
and deciding what you are being asked to find.
We must also remember what we have been discussing the last
several days/weeks with the Pythagorean Theorem, Special
Right Triangles, and Trig Ratios. SOH CAH TOA is vital in these
problems.
An observer on the 1st floor of an airport control tower
sights an airplane at an angle of elevation of 32◦. The
pilot reports the plane’s altitude is 3.5 km. What is the
plane’s horizontal ground distance from the tower?
A helicopter pilot sights a life raft. The angle of
depression is 26° and the helicopter’s altitude is
3km. What is the helicopter’s ground distance from
the raft?
A monument casts a shadow 215 ft long when
the angle of elevation of the sun is 52°. Find
the height of the monument.
The length of a guy wire supporting a radio tower is
175 ft. The angle of elevation created by the guy wire
and ground is 65°. How tall is the tower?
The tailgate of a moving van is 3.5 feet above the
ground. A loading ramp is attached to the rear of
the van at an incline of 10°. Find the length of
the ramp.
Oscar is in a lighthouse on a cliff. He observes 2
sailboats due east of the lighthouse. The angles
of depression to the 2 boats are 33° and 57 °.
Find the distance between the 2 sailboats if the
top of the lighthouse measures 803 feet from sea
level.
A pilot is flying at 10,000 feet and wants to take the
plane up to an altitude of 20,000 feet over the next
50 miles. What should his angle of elevation be to
accomplish this task?
Two observers are 200 feet apart, in line with a tree
containing a bird’s nest. The angles of elevation to the
bird’s nest are 30 ° and 60 °. How far is each observer
from the base of the tree? Is their difference 200?
h
60°
30°
1
2
200
T
x
X=100
Driving along a straight flat stretch of Arizona highway, you spot a
particularly tall saguaro ("suh-WARH-oh") cactus right next to a
mile marker. Watching your odometer, you pull over exactly twotenths of a mile down the road. Retrieving your son's theodolite
from the trunk, you measure the angle of elevation from your
position to the top of the saguaro as 2.4°. Accurate to the nearest
whole number, how tall is the cactus?
You were flying a kite on a bluff, but you managed somehow to dump
your kite into the lake below. You know that you've given out 325 feet
of string. A surveyor tells you that the angle of declination from your
position to the kite is 15°. How high is the bluff where you and the
surveyor are standing?
A lighthouse stands on a hill 100 m above sea level. If ∠ACD measures
60° and ∠BCD is 30°, find the height of the lighthouse.
John wants to measure the height of a tree. He walks exactly 100 feet from the base
of the tree and looks up. The angle from the ground to the top of the tree is 33º .
How tall is the tree?
An airplane is flying at a height of 2 miles above the ground. The distance along the
ground from the airplane to the airport is 5 miles. What is the angle of depression
from the airplane to the airport?
To measure the width of a crater on Mars, the Mar’s Probe travels
at an altitude of 5.3 km above Mar’s surface. The onboard
guidance system measured the angles of depression to the far and
near edges of the crater and found them to be 14° and 23°
respectively. Find the distance across the crater.