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Angle of Depression

angle of depression-1Angle of depression is defined as the angle between the line of sight and the horizontal, when you are looking downwards at an object. Since the angle is formed when you are looking down, an angle of depression is formed.

angle of depression-2

Remember in a right angled triangle, the angle of depression will be the same as the angle of elevation, since the horizontal and the ground are parallel to each other.


Now let us look at some examples in angle of depression:

Example 1: The angle of depression from the top of a vertical cliff 220 m in height to a ship is 28°. How far is the ship from the base of the cliff?

angle of depression-3tan 28°  =  220/x

x  =   220/{tan~28°}

=  413.7598.. m

=  414 m

So the ship is 414 m from the base of the building.


Example 2 : Jason is on top of a 40 m cliff. He observes a boat 800m away from the base of the cliff. Find the angle of depression from Jason to the boat. Answer to the nearest degree.

angle of depression-4The angle of depression ∠DJB = ∠CBJ = θ (alternate angles)

tan θ  =  40/8001/20

θ = tan^-1(1/20)

=  2° 51′ 44″

=  3°

The angle of depression to the boat is 3°.


Example 3 : Steven spots a yacht from the top of a lighthouse L which is 150 m tall. The angle of depression at that instant is 28°. After 2 minutes, he sees the same yacht with an angle of depression of 35°. How far to the nearest metre did the yacht travel in those 2 minutes. Assume sea level to be flat for calculation purposes.

angle of depression-5From Delta{LSH} tan 28°  =  150/SH

SH =  150/{tan 28°}

=  282.1089 m

From DeltaLRH, tan 35°  =  {150/{RH}}

RH =  150/{tan 35°}

=  214.222 m

RS  =  x  =  SH – RH

=  282.1089 – 214.222

=  67.887 m

=  68 m

So the yacht travelled 68 m in those 2 minutes.


Example 4 : From the top of a slide, a child sees a cat. The cat is 20 m away from the foot of the slide. If the angle of depression is 75°, find the height of the slide to 1 decimal place.

angle of depression-6tan 75°  =  x/20

x  =   20tan 75°

=  5.359 m

=  5.4 m

The height of the slide is 5.4 m