Angelevdepr

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

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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

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