29031 Real World Sohcahtoa Worksheet

User Manual: 29031

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I. Model Problems
II. Practice
III. Challenge Problems
IV. Answer Key
Web Resources
SOHCAHTAO
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Real World Applications
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Applications of Right Triangle Trigonometry: Angles of Elevation
and Depression
Preliminary Information: On most maps, it is customary to orient oneself relative to the direction
north: for this reason, north is almost always indicated on every map. Likewise, when working
with real-life trigonometry problems, it is very common to orient angles relative to a horizontal
line.
An angle of elevation refers to the
acute angle a line (or ray, segment,
etc.) makes with a horizontal line,
when measured above the horizontal
(hence an angle of elevation). For
example, the sun’s rays could form
a 23° angle of elevation (above the
horizon).
An angle of depression refers to the
acute angle a line makes with a
horizontal line, when measured
below the horizontal (hence an angle
of depression). For example, an
airplane pilot could look down and
see a feature on the ground below at a 35° angle of depression (below the horizon).
Angles of elevation and depression typically have their vertex at the point where an observer is
positioned. In the previous example, notice that the vertex of the 35° angle is located at the
pilot’s location.
Because horizontal lines are everywhere parallel,
angles of depression and elevation are numerically
equivalent because they form alternate interior
angles of parallel lines:
Students should always be encouraged to consider the following two ideas when they see either
phrase mentioned in a problem:
You may always draw an additional horizontal line on any diagram extending from any
point in the diagram. Just as you did in Geometry, drawing such an auxiliary line can
help to make a complex problem simpler.
The most common error students make when they encounter these terms is they mark an
angle relative to a vertical line (such as an angle with a wall, building, or tree) instead of
with a horizontal line. As stated earlier, always feel free to draw in an auxiliary
horizontal line.
23°
horizontal line
Angles of elevation are measured
up from the horizontal.
35°
horizontal line
Angles of depression are measured
down from the horizontal.
angle of elevation
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Part I) Model Problems
Example 1: Consider right
DFG
pictured at right.
Classify each angle as an angle of elevation, an angle of
depression, or neither.
Step 1: Highlight the horizontal line(s) in the
figure.
Step 2: Determine which acute angles are formed with the horizontal. In this
example,
F
and
G
are formed with the horizontal, but only
G
is acute.
Step 3: Classify each angle:
D
is neither an angle of depression nor an angle of elevation, as
it is formed with vertical line segment
DF
.
F
is formed with a horizontal line segment, but it is not an acute
angle. So it is neither. (In reality, there would be no harm in
specifying it as a 90° angle of elevation, but it is simpler just to say
that
F
is a right angle.
G
is the only acute angle measured from the horizontal; because
line segment
DG
is above the horizontal, it is an angle of
elevation.
Example 2: Michael, whose eyes are six feet off the ground, is standing 36 feet away
from the base of a building, and he looks up at a 50° angle of elevation to a point on the
edge of building’s roof. To the nearest foot, how tall is the building?
Step 1: Make a detailed sketch of the situation.
Make sure to include auxiliary horizontal lines
as needed.
F
G
D
F
G
D
horizontal
6 feet
36 feet
50°
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Step 2: Assign variables to represent the
relevant unknowns. In this example, we shall
use the variable y to represent the vertical leg
of the right triangle, and h to represent the
height of the building.
Step 3: Use SOHCAHTOA to solve for the
unknown side of the right triangle:
fty
y
y
y
90.42
)19175.1(36
50tan36
36
50tan
(Note that units of feet were dropped for simplicity.)
Step 4: Determine the height of the building: Since Michael’s eyes are six feet
from the ground, we must add six feet to variable y to get h:
)(49
90.48
90.426
6
roundedfth
fth
h
yh
Step 5: Check for reasonableness: If Michael were looking up at a 45° angle of
elevation, y would be 36 feet due to the isosceles triangle created. Because he is
looking up at a greater angle, it is reasonable that y is greater than 36 feet.
Adding 6 feet accounts for the fact that his eyes are 6 feet from the ground.
Example 3: A pilot is traveling at a height of 35,000 feet above level ground. According
to her GPS, she is 40 miles away from the airport runway, as measured along the ground.
At what angle of depression will she need to look down to spot the runway ahead?
Step 1: Make a detailed sketch of the situation. Make sure to include auxiliary
horizontal lines as needed.
6 feet
36 feet
50°
y
h
35,000 ft
40 miles
horizontal auxiliary line
unknown angle of depression
runway
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Step 2: Assign variables to represent the relevant unknowns. In this example, we
shall use the variable x to represent the unknown angle of depression:
Because the ground is horizontal, and the auxiliary line is horizontal, we
can properly assume that both angles marked x in the figure are congruent,
as they are both alternate interior angles of parallel lines.
Step 3: Use SOHCAHTOA to solve for the unknown angle:
miles
ft
x40
000,35
tan
Because we have mixed units, we recall that there are 5280 feet in a mile
to convert 40 miles to feet:
200,211
000,35
200,211
000,35
40
000,35
tan 5280
ft
ft
miles
ft
x
mile
ft
To solve for the unknown angle, we use the inverse tangent function:
41.9
200,211
000,35
tan
200,211
000,35
tan
1
x
x
x
Step 4: Check for reasonableness: 40 miles is much larger than 35,000 feet, so it
seems reasonable that the pilot would look down at an angle of only a few
degrees.
35,000 ft
40 miles
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Part II) Practice Problems
1. Classify each of the three angles in the figure at right
as an angle of elevation, an angle of depression, or
neither.
2. Multiple-Choice: A 15 foot ladder rests against a tree on level ground and forms a 75°
angle of elevation. Where is the correct location of the 75° angle?
A) Between the ladder and the ground
B) Between the ladder and the tree
C) Between the tree and the ground
D) It is not possible to place a 75° angle on such a figure.
3. Tammi Jo, whose eyes are five feet off the ground, is standing 50 feet away from the
base of a building, and she looks up at a 73° angle of elevation to a point on the edge of
building’s roof. To the nearest foot, how tall is the building?
4. A pilot is traveling at a height of 30,000 feet above level ground. She looks down at
an angle of depression of 6° and spots the runway. As measured along the ground, how
many miles away is she from the runway? Round to the nearest tenth of a mile.
5. A dog, who is 8 meters from the base of a tree, spots a squirrel in the tree at an angle
of elevation of 40°. What is the direct-line distance between the dog and the squirrel?
A
B
C
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6. A ship is on the surface of the water, and its radar detects a submarine at a distance of
238 feet, at an angle of depression of 23°. How deep underwater is the submarine?
7. The sun is at an angle of elevation of 58°. A tree casts a shadow 20 meters long on
the ground. How tall is the tree?
8. Two observers on the ground are looking up at the top of the same tree from two
different points on the horizontal ground. The first observer, who is 83 feet away from
the base of the tree, looks up at an angle of elevation of 58°. The second observer is
standing only 46 feet from the base of the tree. (Note: you may ignore the heights of the
observers and assume their measurements are made directly from the ground.)
a) How tall is the tree, to the nearest foot?
b) At what angle of elevation must the second observer look up to see the top of
the tree?
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9. Error Analysis: Consider the following problem, which Stephanie and Adam are both
trying to solve:
“A cat, who has climbed a tree, looks down at a dog at a 28° angle of depression.
If the dog is 34 meters from the base of the tree, how high up is the cat?”
The first steps of their work are shown below. Analyze their work and determine who, if
anyone, has set it up correctly.
Stephanie’s work
Adam’s work
x
34
28tan
34
28tan x
10. Complete problem 9: How high up in the tree is the cat?
Part III) Challenge Problems
11. A person starts out 17 miles from the base of a tall mountain, and looks up at a 4°
angle of elevation to the top of the mountain. When they move 12 miles closer to the
base of the mountain, what will be their angle of elevation when they look to the top?
Answer to the nearest degree.
34 meters
28°
34 meters
28°
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12. A pilot maintains an altitude of 25,000 feet over level ground. The pilot observes a
crater on the ground at an angle of depression of 5°. If the plane continues for 16 more
miles, what will be the angle of depression to the crater? Answer to the nearest degree.
13. An observer on the ground looks up to the top of a building at an angle of elevation
of 30°. After moving 50 feet closer, the angle of elevation is now 40°. Consider the
diagram below:
a) Set up an equation representing the situation from the first vantage point.
Your equation will incorporate the 30° angle, x, y, and the 50 feet.
b) Set up an equation representing the situation from the second vantage point.
Your equation will incorporate the 40°, x, and y.
c) You now have two equations in two variables. Solve them simultaneously to
determine the value of x, the distance from the second vantage point to the base of
the building.
30°
40°
50 feet
x
y
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d) Solve for y, the height of the building.
14. Two observers (located at points A and B in the diagram) are watching a climber on
the opposite face of a chasm. The chasm is 81 feet wide. When observer A looks down
to the bottom of the opposite wall of the chasm, he must look down at an angle of
depression of 51°. However, observer A sees the climber at an angle of depression of 20°.
Observer B will see the climber at what angle of elevation?
A
B
81 feet
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Part IV) Answer Key
1. Angle A is an angle of depression; B is neither; C is neither.
2. A
3. 169 feet
4. 54.1 miles
5. 10.4 meters
6. 93 feet
7. 32 feet
8. a) 133 feet
b) 71°
9. Stephanie’s work is incorrect because she drew the 28° angle relative to the vertical
tree. Adam’s work is correct because he drew the 28° angle relative to the horizontal.
Technically, Adam drew an angle of elevation of 28°, but because all horizontal lines are
parallel, his placement of the angle and the equation he wrote will, in fact, yield the
correct solution.
10. The cat has climbed up approximately 18 meters.
11. 13°
12. 7°
13.
a)
x
y
50
30tan
b)
x
y
40tan
c)
ftx
x
x
xx
xx
Therefore
xy
xy
3.110
30tan40tan
30tan50
30tan50)30tan40(tan
30tan5030tan40tan
30tan30tan5040tan
40tan
30tan30tan50
d)
fty
y
xy
5.92
40tan3.110
40tan
14. 41°

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