2. 2
Objectives:
2. Solve real-life word problems involving angles of elevation and angles of
depression.
1. Illustrate and differentiate angles of elevation and angles of depression.
3. 3
Warm Up
Find the exact value of the following:
1. sin 45°
2. cos 30°
3. tan 30°
4. csc 60°
5. sec 45°
6. cot 60°
7. tan 45°
8. sin 30°
𝟐
𝟐
𝟑
𝟐
𝟑
𝟑
𝟐
𝟑
𝟑
𝟐
𝟑
𝟑
𝟏
𝟏
𝟐
4. 4
9. Identify the pairs of alternate interior angles.
2 and 7
10. In the figure below, if the value of 𝒚 = 𝟏𝟐 𝟐:
a. what is the value of x?
b. what is the value of z?
12 𝟐 =
12 𝟔
𝒛 = 𝟏𝟐 𝟔 − 𝟏𝟐 𝟐
3 and 6
Alternate interior angles are
congruent (AIAC).
12 𝟐
6. 6
A line of sight is an imaginary line that connects the eye of an observer to the object
being observed.
7. 7
1 is the angle of elevation from the tower T to the plane P.
An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the
line.
8. 8
An angle of depression is the angle formed by a horizontal line and a line of sight to a point
below the line.
2 is the angle of depression from the plane P to the tower T.
9. 9
Since horizontal lines are parallel, 1 2 by the Alternate Interior Angles Theorem. Therefore,
the angle of elevation from one point is congruent to the angle of depression from the other point.
10. 10
1 is formed by a horizontal line and a line of sight to a point below the line.
Identify each angle as an angle of elevation or
an angle of depression.
∠𝟏 angle of depression.
11. 11
Identify each angle as an angle of elevation or
an angle of depression.
4 is formed by a horizontal line and a line of sight to a point above the line.
∠4 angle of elevation
12. 12
Identify each angle as an angle of elevation or
an angle of depression.
∠𝟐
∠𝟑
angle of elevation
angle of depression
13. 13
Group Activity.
Illustrate the given problems, label the given information then solve.
Show your solution and use x for the indicated unknown part.
14. 14
CRITERIA 5 4 3 2 1
Presentation
Diagram is
accurate,
creative, neat
and
presentable.
Diagram is
accurate,
neat and
presentable.
Diagram is
accurate but
not neat.
Diagram is
neat but not
accurate.
Diagram is
not accurate
and not neat
Accuracy of
Computation
The
computation
is clear. The
answer is
accurate and
with correct
label.
The
computation
is clear. The
answer is
accurate but
with incorrect
label.
The
computation
is not clear.
The answer is
accurate but
with incorrect
label.
The
computation
is clear but
the answer is
inaccurate.
The
computation
is not clear
and the
answer is
inaccurate.
Rubrics
15. 15
1. A plane is flying at an altitude of 12,000 m. From the pilot, the angle of depression to the airport tower is 32º.
How far is the tower from a point directly beneath the plane?
2. A tree 10 meters high casts a 17.3 meter shadow. Find the angle of elevation of the sun.
3. A man flies a kite with a 100 foot string. The angle of elevation of the string is 52 o . How high off the ground is
the kite?
4. From a point on the ground 12 ft from the base of a flagpole the angle of elevation of the top of the pole
measures 53º. How tall is the flagpole?
5. From an airplane at an altitude of 1200 m, the angle of depression to a building on the ground measures 28º.
Find the distance from the plane to the building.
16. 16
Solution: 𝑡𝑎𝑛 32 ° =
12,000
𝑥
1. A plane is flying at an altitude of 12,000 m. From the pilot, the angle of depression to the airport
tower is 32º. How far is the tower from a point directly beneath the plane?
𝑥 𝑡𝑎𝑛 32° = 12,000
𝒙 = 𝟏𝟗, 𝟐𝟎𝟒
𝟏𝟗, 𝟐𝟎𝟒 𝒎
𝟏𝟐, 𝟎𝟎𝟎 𝒎
𝟑𝟐°
𝒙
𝟑𝟐°
𝑡𝑎𝑛 32° 𝑡𝑎𝑛 32°
17. 17
2. A tree 10 meters high casts a 17.3 meter shadow. Find the angle of elevation of the sun.
Solution: tan 𝒙 =
𝟏𝟎
𝟏𝟕.𝟑
𝑥 = 𝑡𝑎𝑛−1
10
17.3
𝒙 = 𝟑𝟎°
𝟑𝟎°
𝟏𝟕. 𝟑 𝒎
𝟏𝟎 𝒎
𝒙
18. 18
3. A man flies a kite with a 100 foot string. The angle of elevation of the string is 52°. How high off the
ground is the kite?
Solution: 𝒔𝒊𝒏 𝟓𝟐° =
𝒙
𝟏𝟎𝟎
𝑥 = 100𝑠𝑖𝑛 52°
𝒙 = 𝟕𝟖. 𝟖𝟎
The kite is 𝟕𝟖. 𝟖𝟎 𝒇𝒕 high off the ground.
𝟏𝟎𝟎 𝒇𝒕
𝒙
𝟓𝟐°
𝟕𝟖. 𝟖𝟎 𝒇𝒕
19. 19
4. From a point on the ground 12 ft from the base of a flagpole, the angle of elevation of the top of the pole
measures 53º. How tall is the flagpole?
Solution: 𝐭𝐚𝐧 𝟓𝟑° =
𝒙
𝟏𝟐
𝑥 = 12 tan 53°
𝒙 = 𝟏𝟓. 𝟗𝟐
𝟏𝟓. 𝟗𝟐 𝒇𝒕
𝟏𝟐 𝒇𝒕
𝟓𝟑°
𝒙
20. 20
5. From an airplane at an altitude of 1200 m, the angle of depression to a building on the ground measures 28º.
Find the distance of the plane from the building.
Solution: sin 28° =
1200
𝑥
1,200 𝑚
𝒙 = 𝟐𝟓𝟓𝟔 𝒎
𝟐𝟓𝟓𝟔 𝒎
28°
𝑥 𝑠𝑖𝑛 28° = 1200
𝒙
28°
𝑠𝑖𝑛 28° 𝑠𝑖𝑛 28°
21. 21
Remember:
A line of sight is an
imaginary line that
connects the eye of an
observer to the object being
observed.
An angle of elevation is the
angle formed by a
horizontal line and a line of
sight to a point above the
line.
An angle of depression is
the angle formed by a
horizontal line and a line of
sight to a point below the
line.
22. 22
A. Identify each angle as an angle of elevation or an angle of depression.
B. How far from the door must a ramp begin in
order to rise 3 feet with an 8° angle of elevation?
Solution: tan 8° =
3
𝑥
𝑥 =
3
tan 8°
𝑥 = 21.35 𝑓𝑡
Angle of elevation
Angle of depression
Angle of elevation
𝒙
3 𝑓𝑡
𝟖°
23. 23
1. The angle of elevation from a boat to the top of a 92-meter hill is 12º.How far is the boat from the base of
the hill?
2. From the top of a control tower 250 m tall, an airplane is sighted on the ground below. If the airplane is 170
m from the base of the tower, find the angle of depression of the airplane from the top of the control tower.
3. A 12-meter high post casts a 19-meter shadow. Find the angle of elevation to the sun.
Illustrate the given problems, label the given information then solve. Show your solution and use x
for the indicated unknown part.
Assignment 3.
