Angle of Elevation/Angle of Depression Examples

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The Mathematics 11
Competency Test
Angle of Elevation/Angle of
Depression Examples
One of the classic examples in right triangle trigonometry is the calculation of the so-called angles
of elevation or depression:
•
When a person looks at something above his or her location, the angle between the line
of sight and the horizontal is called the angle of elevation. In this case, the line of sight
is “elevated” above the horizontal.
•
When a person looks at something below his or her location, the angle between the line
of sight and the horizontal is called the angle of depression. In this case, the line of
sight is “depressed” below the horizontal.
Since the vertical and horizontal directions are perpendicular, the elements of problems dealing
with the relationship between lines of sight and the horizontal lead naturally to right triangles:
horizontal
angle of
depression
angle of
elevation
horizontal
Here are some examples illustrating these ideas.
Example: A person stands at the window of a building so that his eyes are 12.6 m above the
level ground in the vicinity of the building. An object is 58.5 m away from the building on a line
directly beneath the person. Compute the angle of depression of the person’s line of sight to the
object on the ground.
Solution: The angle of depression of the line of
sight is the angle, θ, that the line of sight makes
with the horizontal, as shown in the figure to the
right. Since the ground is level, it is parallel to any
horizontal line, and so the angle that the line of
sight makes with the ground is equal to θ as well.
As a result, we have
tanθ =
12.6
58.5
horizontal
θ
ground
θ
58.5 m
so that
 12.6 
−1  12.6 
0
 = tan  58.5  ≅ 12.15
58.5




θ = arctan 
Here we’ve shown two commonly used notations, arctan and tan-1, for the inverse tangent
function. Thus, the angle of depression of the line of sight to the object is 12.150 rounded to two
decimal places.
David W. Sabo (2003)
Angle of Elevation/Angle of
Depression
Page 1 of 2
Example: Calculate the angle of elevation of the line of sight of a person whose eye is 1.7 m
above the ground, and is looking at the top of a tree which is 27.5 m away on level ground and
18.6 m high.
Solution: The angle of elevation
is the angle the line of sight
makes with the horizontal when
the line of sight is upwards or
above the horizontal (in contrast
to the situation where we use the
term “angle of depression” to
refer to a line of sight which is
downwards, or below the
eye level
θ
horizontal). So, schematically,
the situation here is as shown in
1.7 m
the figure to the right, with the
27.5 m
symbol θ indicating the required
person
tree
angle of elevation. Note that the
right triangle for which the line of
sight forms the hypotenuse is
16.9 m high after we take into account the 1.7 m distance that the observer’s eye is above the
ground. Thus
tanθ =
16.9
27.5
so that
 16.9 
0
 ≅ 31.57
27.5


θ = tan−1 
Thus, to two decimal places, the angle of elevation of this person’s line of sight is 31.570.
David W. Sabo (2003)
Angle of Elevation/Angle of
Depression
Page 2 of 2
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