The document explains the concepts of angles of elevation and depression, detailing their definitions and providing examples of how to solve related problems. It includes various scenarios such as the height of the Seattle Space Needle and distances related to observers at different heights looking at objects below or above. Additionally, it outlines the use of trigonometric functions to calculate these distances with specific angles.

1. ANGLES OF
ELEVATION &
DEPRESSION

2. Solve problems involving angles
of elevation and angles of
depression.
You will

3. angle of elevation
angle of depression
Vocabulary

4. An angle of elevation is the angle formed by a
horizontal line and a line of sight to a point above the
line. In the diagram, 1 is the angle of elevation
from the tower T to the plane P.
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 to
the tower.

5. 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.

6. Classify each angle as an angle
of elevation or an angle of
depression.
1
1 is formed by a horizontal line and a line of sight to a point
below the line. It is an angle of depression.
4
4 is formed by a horizontal line and a line of sight to a point
above the line. It is an angle of elevation.
2
2 is formed by a horizontal line and a line of sight to a point
above the line. It is an angle of elevation.

7. Use the diagram above to
classify each angle as an angle
of elevation or angle of
depression.
5
6
6 is formed by a horizontal line and a line of sight to a
point above the line. It is an angle of elevation.
5 is formed by a horizontal line and a line of sight to a
point below the line. It is an angle of depression.

8. ∠FL
S
∠LS
T
∠RT
W
∠TW
S
∠DC
B
∠AB
C
∠WZ
P
∠ZP
R

9. ∠FL
S
∠LS
T
∠RT
W
∠TW
S
∠DC
B
∠AB
C
∠WZ
P
∠ZP
R

10. The Seattle Space Needle casts a 67-meter shadow. If the
angle of elevation from the tip of the shadow to the top of the
Space Needle is 70º, how tall is the Space Needle? Round to
the nearest meter.
B
A 67 m
y
70o
tan 70 °=
𝑦
67
y =67 tan 70 °
𝑦 ≈ 184 𝑚

11. At the circus, a person in the audience at ground level watches the high-
wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is
25 feet off the ground. How far is the audience member from the base of
the platform, if the angle of elevation from the audience member’s line
of sight to the top of the acrobat is 27°?
Make a drawing.

12. Maria is at the top of a cliff and sees a seal in the water. If the
cliff is 40 feet above the water and the angle of depression is
52°, what is the horizontal distance from the seal to the cliff,
to the nearest foot?
𝑥 ¿
40
tan 52 °
tan 52° =
40
𝑥
𝑥 ≈ 31.3 𝑓𝑡 .

13. A plane is at an altitude of 3500 feet and the angle of
elevation from the airport to the plane is 29°. What is the
horizontal distance between the plane and the airport?
Round to the nearest foot.
3500 ft
29°
𝑥≈6314 𝑓𝑡 .
tan 29°=
3500
𝑥
𝑥 ¿
3500
tan 29 °

14. An ice climber stands at the edge of a crevasse that is 115
ft. wide. The angle of depression from the edge where she
stands to the bottom of the opposite side is 52º. How deep
is the crevasse at this point? Round to the nearest foot.
Draw a sketch to represent the given information. Let C
represent the ice climber and let B represent the bottom of the
opposite side of the crevasse. Let y be the depth of the crevasse.
𝑦 ¿(115) tan52°
tan 52° =
𝑦
115
𝑦 ≈ 147 𝑓𝑡 .

15. Vernon is on the top deck of a cruise ship and observes two dolphins
following each other directly away from the ship in a straight line.
Vernon’s position is 154 meters above sea level, and the angles of
depression to the two dolphins are 35° and 36°. Find the distance
between the two dolphins to the nearest meter.
219.93 – 211.96 = 7.97 m

16. A ranger sees another fire and the angle of depression to the fire
is 3°. What is the horizontal distance to this fire? Round to the
nearest foot.
3°
𝑥=
90
tan 3 °
tan 3 °=
90
𝑥
𝑥 ≈ 1717 𝑓𝑡 .

17. An observer in a lighthouse is 69 ft above the water. He
sights two boats in the water directly in front of him. The
angle of depression to the nearest boat is 48º. The angle of
depression to the other boat is 22º. What is the distance
between the two boats? Round to the nearest foot.
tan 48 °=
69
𝑦
y ¿
69
tan 48 °
≈ 62.1 𝑓𝑡
tan 22°=
69
𝑧
z ¿
69
tan 22 °
≈ 170.8 𝑓𝑡
𝟏𝟕𝟎.𝟖−𝟔𝟐.𝟏≈𝟏𝟎𝟗 𝒇𝒕

18. A. 37°
B. 35°
C. 40°
D. 50°
At a diving competition, a 6-foot-tall diver stands atop the 32-foot
platform. The front edge of the platform projects 5 feet beyond the ends
of the pool. The pool itself is 50 feet in length. A camera is set up at the
opposite end of the pool even with the pool’s edge. If the camera is
angled so that its line of sight extends to the top of the diver’s head,
what is the camera’s angle of elevation to the nearest degree?

19. A. 19 ft
B. 20 ft
C. 44 ft
D. 58 ft
Luisa is in a hot air balloon 30 feet above the ground. She
sees the landing spot at an angle of depression of 34.
What is the horizontal distance between the hot air
balloon and the landing spot to the nearest foot?

20. A. 14 ft
B. 24 ft
C. 37 ft
D. 49 ft
Madison looks out her second-floor window, which is 15 feet above the
ground. She observes two parked cars. One car is parked along the
curb directly in front of her window and the other car is parked directly
across the street from the first car. The angles of depression of
Madison’s line of sight to the cars are 17° and 31°. Find the distance
between the two cars to the nearest foot.