Right Triangle Trigonometry

1
Right Triangle Trigonometry

• By Jeffrey Bivin
• Lake Zurich High School
• jeff.bivin_at_lz95.org

Last Updated December 9, 2010
2
Review of Previous Applications
3
A surveyor is standing 45 feet from the base of a
large tree. The surveyor measures the angle of
elevation from the ground to the top of the tree
to be 67.5o. Find the height of the tree.
h
67.5o
45 feet
4
An airplane flying at 4500 feet is on a flight
path directly toward an observer. If 30o is the
angle of elevation from the observer to the
plane, find the distance from the observer to the
plane.
d
4500 feet
30o
5
In traveling across flat land a driver noticed a
mountain directly in front of the car. The angle
of elevation to the peek is 4o. After the driver
traveled 10 miles, the angle of elevation was
11o. Approximate the height of the mountain.
h
11o
4o
10 mi
x
10 x
6
A flagpole at the top of a tall building (and at
the edge of the building) is know to be 45 feet
tall. If a man standing down the street from the
building calculates the angle of elevation to the
top of the building to be 55o and the angle of
elevation to the top of the flagpole to be 57o.
Find the height of the building.
45
h
57o
55o
d
7
An observer standing on the cliff adjacent to the
ocean looks out and sees an airplane flying
directly over a ship. The observer calculates
the angle of elevation to the plane to be 14o and
the angle of depression to the ship to be 27o.
How high above the ship is the airplane if we
know that the ship is 1.5 miles from shore?
p
14o
1.5 mi
27o
b
Distance of plane above ship
8
In Washington, D.C., the Washington Monument is
situated between the Capitol and the Lincoln
Memorial. A tourist standing at the Lincoln
Memorial tilts her head at an angle of 7.491 in
order to look up to the top of the Washington
Monument. At the same time, another tourist
standing at the Capitol steps tilts his head at a
5.463 to also look at the top of the Washington
Monument. Find the distance from the Lincoln
Memorial to the Washington Monument.
9
Additional Applications
10
A ship leaves port at noon and heads due west at
20 knots, or 20 nautical miles (nm) per hour. At
2 P.M. the ship changes course to N 54o W. Find
the ships bearing and distance from the port of
departure at 3 P.M.
d
a
54o
20sin(36o)
20 nm
78.181o
20 nmph for 1 hr
36o
?
b
40 nm
20cos(36o)
20 nmph for 2 hrs
Bearing N 78.181o W
11
A ship leaves port at noon and heads due west at
20 knots, or 20 nautical miles (nm) per hour. At
2 P.M. the ship changes course to N 54o W. Find
the ships bearing and distance from the port of
departure at 3 P.M.
d
a
54o
20sin(36o)
20 nm
78.181o
20 nmph for 1 hr
36o
?
b
40 nm
20cos(36o)
20 nmph for 2 hrs
Bearing N 78.181o W
12
Two lookout towers are 50 kilometers apart.
Tower A is due west of tower B. A roadway
connects the two towers. A dinosaur is spotted
from each of the towers. The bearing of the
dinosaur from A is E 47o N. The bearing of the
dinosaur from tower B is W 32o N. Find the
distance of the dinosaur to the roadway that
connects the two towers.
h
47o
32o
A
B
x
50 x
13
h
47o
32o
A
B
x
50 x
19.741 km
14
Two lookout towers spot a fire at the same time.
Tower B is Northwest of Tower A. The bearing of
the fire from tower A is N 33o W and is
calculated to be 45 km from the tower. The
bearing of the fire from tower B is N 63o E and
is calculated to be 72 km from the tower. Find
the distance between the two towers and the
bearing from tower A to tower B.
a
c
63o
72
d
B
b
45
s
33o
b d
45cos(330) 72sin(630)
A
a c
45sin(330) 72sin(630)
15
a
c
63o
72
d
B
b
45
s
88.805 km
33o
b d
45cos(330) 72sin(630)
A
a c
45sin(330) 72sin(630)
16
a
c
63o
72
d
B
b
45
s
88.805 km
33o
b d
45cos(330) 72sin(630)
?
A
a c
45sin(330) 72sin(630)