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David W. Sabo (2003) Angle of Elevation/Angle ofDepression

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horizontalangle ofelevation

horizontal

angle ofdepression

58.5 m

horizontal

ground

The Mathematics 11Competency Test Angle of Elevation/Angle of

Depression Examples

One of the classic examples in right triangle trigonometry is the calculation of the so-called anglesof elevation or depression:

When a person looks at something above his or her location, the angle between the lineof sight and the horizontal is called the angle of elevation. In this case, the line of sightis elevated above the horizontal.

When a person looks at something below his or her location, the angle between the lineof sight and the horizontal is called the angle of depression. In this case, the line ofsight is depressed below the horizontal.

Since the vertical and horizontal directions are perpendicular, the elements of problems dealingwith the relationship between lines of sight and the horizontal lead naturally to right triangles:

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 thelevel 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 persons line of sight to theobject on the ground.

Solution: The angle of depression of the line of

sight is the angle, , that the line of sight makeswith the horizontal, as shown in the figure to theright. Since the ground is level, it is parallel to anyhorizontal line, and so the angle that the line of

sight makes with the ground is equal to as well.As a result, we have

12.6tan

58.5 =

so that

1 012.6 12.6arctan tan 12.1558.5 58.5

= =

Here weve shown two commonly used notations, arctan and tan-1

, for the inverse tangentfunction. Thus, the angle of depression of the line of sight to the object is 12.15

0rounded to two

decimal places.

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David W. Sabo (2003) Angle of Elevation/Angle ofDepression

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1.7 m

27.5 m

eye level

person tree

Example: Calculate the angle of elevation of the line of sight of a person whose eye is 1.7 mabove the ground, and is looking at the top of a tree which is 27.5 m away on level ground and18.6 m high.

Solution: The angle of elevationis the angle the line of sightmakes with the horizontal whenthe line of sight is upwards orabove the horizontal (in contrastto the situation where we use theterm angle of depression torefer to a line of sight which isdownwards, or below thehorizontal). So, schematically,the situation here is as shown inthe figure to the right, with the

symbol indicating the requiredangle of elevation. Note that theright triangle for which the line of

sight forms the hypotenuse is16.9 m high after we take into account the 1.7 m distance that the observers eye is above theground. Thus

16.9tan

27.5 =

so that

1 016.9tan 31.5727.5

=

Thus, to two decimal places, the angle of elevation of this persons line of sight is 31.570.