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Angle of Elevation

This is the angle between normal eye level(horizontal line) and the line through which an object is above the observer’s eye

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This is the angle between the horizontal line and the line through which the object is below the observer’s eye can be seen.

Angle of Depression

This is the angle between the horizontal line and the line through which the object is below the observer’s eye can be seen.

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The angles of elevation and depression are mostly applicable in solving problems involving heights and distances.

Example 1

1. A man is at a point 18m away from the foot of a tree. From that point, the angle of elevation of the top of the tree is 29°. Calculate the height of the tree.

Solution

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Tan P = \( \frac{opp}{adj} \)

∠P = 29°
opp =h
adj = 18

Tan 29° = \( \frac{h}{18} \)

h = \( \scriptsize 18 \: \times \: tan \: 29^o \)

h = \( \scriptsize 18 \: \times \: tan \: 0.5543 \)

h = \( \scriptsize 9.977m \)

Example 2

2. The angle of depression from the top of a building of height 30m of a stationary car is 41°. Find the distance between the car and the top of the building.

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Solution:

Sin C = \( \frac{opp}{hyp} \)

∠C = 41°
opp = 30
hyp = AC

Sin 41° = \( \frac{30}{AC} \)

AC = \( \frac{30}{Sin 41} \)

AC = \( \frac{30}{0.6561} \)

AC= 45.7m

Note: the angle of depression is ∠DAC, therefore ∠BCA is alternate to ∠DAC

Exercise 

1. A boy observes that the angle of elevation of the top of a tower is 32°. He then walks 8m towards the tower and then discovers that the angle of elevation is 43°. Find the height of the tower.

2. From a window 10m above the level ground, the angle of depression of an object on the ground is 25.4°. Calculate the distance of the object from the foot of the building.

3. A boy standing 70m away from a flag-post observes that the angles of elevation of the top and bottom of a tower on top of the flag-post are 57° and 69° respectively. Find the height of the tower.

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