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Geometry
Topic 6
Right Triangle Trigonometry
PROBLEM PACKET
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Homework Assignments!!
Assignment
Number
Topic 6 Problem Packet
Description of Assignment
Date
Assigned
Date
Due
2
6-1 Introduction to Trigonometry
In a right triangle, the side opposite the right angle has a special name, the hypotenuse.
The other sides of the right triangle are called the legs. These two legs are identified as being
opposite or adjacent to each of the acute angles of the triangle.
Example:
In ∆ABC,
Side AC is the hypotenuse.
Leg AB is opposite C and adjacent to A.
Leg BC is opposite A and adjacent to C.
Questions #1 to 5 refer to ∆FUN.
1. The hypotenuse is ______ .
2. The leg opposite F is _____.
3. The leg opposite U is _____.
4. The leg adjacent to F is _____.
5. The leg adjacent to U is _____.
Questions #11 - #15 refer to ∆EVN
11. The hypotenuse is ______.
12. The leg opposite E is _____.
13. The leg opposite N is _____.
14. The leg adjacent to E is _____.
15. The leg adjacent to  N is _____.
Topic 6 Problem Packet
Questions #6 - #10 refer to ∆CMP.
6. The hypotenuse is ______.
7. The leg opposite M is _____.
8. The leg opposite P is _____.
9. The leg adjacent to M is _____.
10. The leg adjacent to P is _____.
In questions #16-18, use ∆ABC and
∆DEF to the left.
16. Why are ABC and DEF similar?
17. Fill in the blanks in the following
ratios:
BC
?
a.
=
AB DE
BC AC
b.
=
EF
?
18. Given that AB = 3, DE = 5 and EF =
6, find BC.
3
Example:
Find sin A, cos A, and tan A. Leave your answers as reduced fractions.
Solution:
First, identify hypotenuse, opposite leg, and adjacent leg.
Remember, opposite and adjacent legs change depending on
which angle you are using.
A
24
C
26
10
Hypotenuse = 26, adjacent = 24, opposite = 10
opp 10 5
sin A =
=
=
hyp 26 13
adj 24 12
cos A =
=
=
hyp 26 13
opp 10 5
tan A =
=
=
adj 24 12
B
Use the diagrams below to find the following trig ratios. Give each answer as a reduced
fraction.
A
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
sin A
cos A
tan A
sin T
cos T
tan T
sin G
cos G
tan G
sin M
cos M
tan M
31.
What does SOHCAHTOA stand for? Why is it helpful to remember?
Topic 6 Problem Packet
17
15
U
7
C
8
T
M
24
25
G
4
Choose the correct acute angle to
complete each trigonometric equation.
12
32.
tan ?
=
35
12
33.
sin ?
=
37
A 12 N
12
34.
cos
?
=
37
35
37
35
35.
sin ?
=
37
35
R
36.
tan ?
=
12
35
37.
cos
?
=
37
Choose the correct trig function (sin,
cos, or tan) to complete each equation.
38.
?
39.
?
40.
?
41.
?
42.
?
8
10
8
E =
6
6
N =
8
6
N =
10
6
E =
10
N =
10
N
E
6
8
W
Topic 6 Problem Packet
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6-2 Finding Missing Sides
Example:
Set up an equation and solve for the indicated side.
Solution:
First, identify which sides you are using with respect to the given angle.
Here, 26 is the hypotenuse, and the missing side x is the opposite leg.
A
32°
C
26
x
B
Then choose the trig function that uses those two sides. Here, we will use sin
32° because sine is opp/hyp.
x
sin 32° =
26
Replace sin 32° with its decimal equivalent found on your trig table or your
calculator:
x
0.5299 =
26
Then solve for x by cross multiplying:
x = 13.78
Use the appropriate trig function to solve for the missing side in each right triangle. Show
all of your work.
1.
2.
3.
70˚ x
17
x
14˚
62˚
14
x
4.
5.
x
C
a
6.
22
55˚
23˚
7.
25
9
A
66
10
x
40
64˚
8.
9.
C
6
B
Topic 6 B
Problem Packet
x
b
48
A
A
55
c
C
1
B
6
10.
11.
13.
14.
16.
17.
12.
15.
For #18-20, find all three variables in each diagram.
18.
8
19.
20.
f
30°
b
37°
10
d
e
c
h
12
43°
a
i
g
Find the indicated ratio using a calculator or your table. Round to 4 decimal places.
21.
sin 55°
Topic 6 Problem Packet
22. tan 82°
23.
cos 21°
24. sin 90°
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6-3 Finding Missing Angles
Example:
Set up an equation and solve for the indicated angle.
Solution:
First, identify which sides you are using with respect to the missing angle.
Here, 26 is the hypotenuse, and 13 is the adjacent leg to B.
A
26
Then choose the trig function that uses those two sides. Here, we will use cos x
because cos is adj/hyp.
13
cos x =
26
13
Find the decimal equivalent for
= .5
26
Look for this decimal in the cosine column of your trig table, or if using your
calculator, use cos-1 (using the 2nd button) and then the decimal.
cos-10.5 = 60°
x
C
B
13
Use the appropriate trig function to solve for the A in each right triangle. Show all of
your work. Round your answer to the nearest degree.
1.
2.
6.3
A
3.
A
C
9
4
4.9
4.
6.
8
45
A
C
B
C
50
2.6
5
A
B
A
For problems #7-15, find the indicated side or angle measure:
A
7. A
8. D
D
4.0
B
2.5
9. G
E
H
3.3
4.6
Topic 6 Problem Packet
F
C
3.6
4.4
C
7.2
B
5.
B
A
4.5
B
C
B
C
I
9.2
G
8
O
10.
11.
O
DG
12.
T
O
12
D
13.
5
U
20
8
M
ST
72
37
G
S
N
VW
X
14.
P
P
21
15.
3.5
Q
CA
A
C
15
2
4.2
54
T
V
W
R
Use the diagram below to find each of the following:
(Hint: You only have to use trig to find one of the angles.)
K
16.
mL
17.
mJ
2.4
1.8
18.
JL
L
J
Use your table or calculator to find the angle measure that would have the indicated trig
value:
19.
cos x° = .8988
20. sin x° = .2079
21. tan x° = 2.6051 22. sin x° = .5000
Find the missing angle in each diagram.
23.
27.
24.
Find mT
Topic 6 Problem Packet
25.
28. Find mD
26.
29. Find x, y, and mB
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6-4 Angles of Elevation and Depression
Example:
A person is standing 100 m from the base of a cell phone tower. The angle of elevation to the
top of the tower is 40°. How tall is the tower, to the nearest tenth of a meter?
First, draw and label a diagram.
Then set up an equation to find x.
Because x is the opposite leg and 100 is the adjacent leg,
use tangent:
x
40°
°
tan 40° =
x
100
.8391 =
x
100
x = 83.9 m
100 m
For each question below:
• Draw a picture that represents the situation.
• Choose the trig ratio that you will need to use.
• Write an equation using the trig ratio that you chose.
• Solve the equation and answer the question.
1. A flagpole casts a shodow 35 feet long. The angle that the sun makes with the ground is 27°.
How tall is the flagpole?
2. Camila is flying a kite with 200 meters of string out. Her kite string makes an angle of 54°
with the level ground. How high is her kite?
3. The entrance to a school is 10 feet above the sidewalk. A wheelchair accesible ramp needs to
start from the sidewalk and go up to the entrance of the school. The ramp must incline at an
angle of 5 degrees. How long must the ramp be?
4. It is believed that Galileo used the ‘Leaning Tower of Pisa’ to conduct his experiments on the
laws of gravity. When he dropped objects from the top of the 55 meter high tower, they
landed 4.8 meters from the base of the tower. What is the angle that the tower leans off from
the ground?
5. From a 40-foot lighthouse, a Coast Guard officer sees a boat sinking. The angle of
depression of the boat is 4. How far is the boat from the lighthouse?
6. A ramp 20 feet in length rises to a loading platform that is 3
1
feet off of the ground. What
2
is the angle of elevation of the ramp?
Topic 6 Problem Packet
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7. A pilot spots a bird at the end of the runway as he begins landing the plane. If the plane’s
path of descent is 2.7 miles long and the plane’s angle of depression is 43, how long is the
runway?
8. A group of students were using shadows to find the heights of trees near their school. They
used the diagram shown here to represent the general situation.

S
In this diagram,  (the Greek letter “theta”) represents the angle of elevation of the sun, and S
represents the length of the tree’s shadow.
a. In one case, they found  = 35° and S = 50 ft. What is the height of the tree?
b. Later that day, with a different tree, they got
height of that tree?
 = 60° and S = 20 feet. What is the
c. Develop a general expression for the height of a tree in terms of S and .
9.
a. A sailboat is in trouble and the
Lookout station
people on board are considering
swimming to shore. A lookout
35°
station on the shore is able to tell
them that they are 2.3 miles from
shoreline
the station and that the line from
the station to the boat forms an
angle of 35° with the shoreline.
(Assume the shoreline is straight,
as shown in this diagram.) If the
Location of sailboat
people are capable of swimming
1.5 miles, will they be able to make it to shore or should they call for help? Explain.
b. The lookout station officer would like to be able to tell people in such situations their
actual distance from shore. Find a general formula the officer can use to find this
distance. Your formula should express this distance in terms of the distance from the
station to the boat and the angle between the line to the boat and the shoreline.
10. Sarah is standing on the top of a building looking through a telescope that is exactly 425 feet
above the ground. Through the telescope she spots a penny on the sidewalk. If the penny is
exactly 911 feet away from the base of the building, what is the telescope’s angle of
depression?
Topic 6 Problem Packet
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11. John is flying a kite. He has let the kite out until he’s holding onto the very end of the string.
The string is 500 yards long. John attaches the string to the ground and the kite keeps flying.
Using a protractor, he determines that the string forms a 55° angle of elevation. How high is
the kite above the ground?
12. A submarine is 626 meters below the surface of the water. While traveling forward, it begins
making an ascent to the surface so that it will emerge on the surface after traveling 4420
meters (diagonally) from the point of its initial ascent.
a.
What angle of elevation did the submarine make?
b.
What was the horizontal distance traveled during the sub’s ascent?
13. Nathan pilots a small plane on weekends. During a recent flight, he determined that he was
flying at an altitude of 3000 feet above the ground and that the ground distance to the start of
the landing strip was 8000 feet.
a.
What is Nathan’s angle of depression to the start of the landing strip?
b.
What is the diagonal distance between the plane in the air and the start of the
landing strip?
14.
The angle of elevation of a wheelchair ramp is 5°, and the ramp is 12ft long. How high is
the top of the ramp from the ground?
15.
A flagpole casts a shadow 4.6 m long. The angle of elevation of the sun is 49°. How tall is
the flagpole?
16.
The top of a lighthouse is 100 m above sea level. The angle of depression down to the
fishing boat is 18°. How far from the base of the lighthouse is the fishing boat?
17.
The angle of depression from the top of an observation tower to a historical statue is 23°.
The statue is 97 m away from the base of the tower. How tall is the tower?
18.
A pillar is 13 m tall. If the angle of elevation of the sun is 55°, how long is the shadow cast
by the pillar?
19.
A 25 ft ladder is leaning against a building. The ladder makes an angle of 65° with the
ground. How far away from the building is the base of the ladder?
20.
Standing on the observation deck of a skyscraper, Superman (who is 6 ft tall) looks down
and sees a man stealing a woman’s purse. If the angle of depression is 54° and the building
is 200 ft high, how far will Superman have to fly to rescue the woman’s purse?
21.
Standing at the top of a hill, a person observes a lake down below. If the angle of
depression to the shore is 62° and the elevation of the hill is 490 ft, what is the horizontal
distance from the shore to the base of the hill if the person is 6 ft tall?
Topic 6 Problem Packet
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