OVERVIEW
Use Trigonometry & Pythagorean Theorem to Solve
G.SRT.8
G.SRT.8
Use trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems.
No surprises here…. Use trigonometry
and the Pythagorean Theorem to
solve problems.
Apply these concepts to solve
problems.
(1) The student will be able interpret
the word descriptions into lengths and
angles of a right triangle so that they
can diagram the relationship.
(2) The student will be able to solve
trigonometry and Pythagorean
Theorem problems based on written
descriptions.
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The emphasis here is on the use of
these two very powerful relationships.
Students need to see a variety of
different situations and be able to
simplify them into a trigonometry or
Pythagorean problem.
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1 – Vocabulary…. Lots of the issues
with these problems are the use of
‘new’ vocabulary - the foot of the
ladder, the angle of elevation, the
altitude of the plane, and so on. Help
students gain a good understanding of
the ‘typical’ questions.
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NOTES
Use Trigonometry & Pythagorean Theorem to Solve
G.SRT.8
CONCEPT 1 – Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems. *(Modeling Standard)
In this objective the principles that have been learned concerning similarity, trigonometry and the Pythagorean Theorem
are applied to solve ‘real’ world problems. While they are called real world problems many of them have been greatly
simplified so that the mathematics at this level can be applied. Real world problems have many variables and are often
too complex to approach with this simple level of mathematics but as students’ progress through mathematics more
detail is added to the problems making them ‘more real’. At this stage students’ must gain proficiency in the skill within
a context to see where these mathematical tools apply.
These real world problems are often called word problems by students. Yes, that is often true… Real world problems do
come with context and so, yes, they look and feel different than the standard symbol based problems that are often
done. It also shows the need to gain proficiency with language as well as mathematics to succeed with these types of
problems. Most of these notes will be discussing the context of these problems and the translation of word to
mathematics and not the solving of them. That skill was discussed in an earlier objective. Start with how angles are
referred to.
THE ANGLE OF ELEVATION
To elevate is the move in an upward direction. Thus the angle of elevation is
an upward angle from the horizon. The angle of elevation is the reference
angle from the position in an upward direction from the horizon.
The stranded person on the island looks up
to the plane at an ANGLE OF ELEVATION.
Very few students struggle with this concept – The name and the placement seem to come easily. This is not true
concerning the next angle, many students struggle with its placement and understanding.
THE ANGLE OF DEPRESSION
To be depressed is to be down, thus the angle of depression is the reference angle from the position, in a downward
direction, from the horizon. Sounds simple, like the angle of elevation but it isn’t quite as easy.
Example #1
Example #2
Example #3
In example #1, the angle of depression (1) has been placed correctly, as the angle from the airplanes’ horizon looking
down. In example #2, the most common error has taken place; the angle of depression (1) was PLACED IN THE WRONG
LOCATION. Student’s make this error because no horizon has been drawn and so they see this as the only angle from
that reference point. Finally, in example #3, the angle of depression (1) has been correctly drawn and located - placing it
from the horizon down.
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NOTES
Use Trigonometry & Pythagorean Theorem to Solve
G.SRT.8
Level 2 Difficulty -- THE USE OF TWO TRIANGLES
The use of two triangles just simply means that two different values must be found to gain the final answer. Really this
problem isn’t an increase of difficulty it is simply an increase of work. It often gains difficulty because it does require
more contexts to explain what is going on.
Example Problem
A flagpole is at the top of a building. 300 ft from the base of the building,
the angle of elevation of the top of the pole is 32 and the angle of elevation
of the bottom of the pole is 30. Determine the length of the flagpole (to
the nearest foot).
This problem looks quite complex because of the overlapping angles
of elevation but really if we think through the problem we can
simplify it by solving two different triangles.
32°
30°
300 ft
The way to get the height of just the
flagpole would be to calculate the
total height from ground to top of flag
(x) and then calculate the distance
from ground to the bottom of the
flagpole (y) and then subtract the two
values.
x
y
32°
30°
300 ft
x
tan 32 
300
x  (tan 32)(300)
x  187.46 ft
300 ft
y
300
y  (tan 30)(300)
y  173.21 ft
tan 30 
( x – y) will gives the flagpole height, 187.46 – 173.21 = 14.25 ft.
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ASSESSMENT
Use Trigonometry & Pythagorean Theorem to Solve
G.SRT.8
1. The angle of elevation from the car to the girl is:
A) 1
B) 2
C) 3
D) 4
2. The angle of depression from the helicopter to the girl is:
A) 1
B) 2
C) 3
D) 4
3. Tommy has caught his kite at the top of a 16 ft tree. From where
Tommy is standing the elevation to the top of the tree is 29, what is the
length of string (round to the nearest foot)?
A) 33 ft
B) 29 ft
C) 18 ft
D) 8 ft
4. A plane at an altitude of 7000 ft is flying in the direction of an island. If angle
of depression is 21 from the plane to the island, what is the horizontal distance
until the plane flies over the island? (round to the nearest foot)
A) 2,687 ft
B) 7,498 ft
C) 18,236 ft
D) 19,533 ft
5. A 18 ft tree casts a 15 ft long shadow. What is the angle formed by the sun’s rays and the
ground? (round to the nearest degree)
A) 60
B) 50 
C) 40 
6. The angle of elevation from the car to the girl
at the top of the parking garage is 11 & 12.
7. The angle of depression from the helicopter
to the boy on the third floor of the parking
garage is 2.
8. The angle of depression from the boy on the
third floor of the parking garage to the car is
10.
D) 30
T or F
3
2
1
4
T or F
5
6
7
8
T or F
9
10
12
11
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ASSESSMENT
Use Trigonometry & Pythagorean Theorem to Solve
G.SRT.8
9. After reading that “the angle of depression from the girl to the car is 55”, Tim
labeled the diagram as shown. Has he labeled the diagram correctly? Explain.
55°
35°
10. Jazmin is flying a kite but it got caught at the top of a 24 ft tree. If she
let out 100 ft of string, what is the angle of elevation from Jazmin to the
top of the tree (round to the nearest degree)?
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ASSESSMENT
Use Trigonometry & Pythagorean Theorem to Solve
G.SRT.8
Answers:
1)
2)
3)
4)
5)
6)
7)
8)
9)
C
C
A
C
B
T
F
F
No, he hasn’t… the angle of depression is from the horizon down. He has placed the 55 between the
building and the line of sight. Instead he should have put it between the girl’s horizon and the line of
sight to the car.
10) sin 1 (24 /100)    13.89  14
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