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Angle of Elevation

An angle of elevation is the angle between the line of sight (an imaginary line from an observer's eye to a perceived object) and the horizontal when an observer looks upward. Suppose that from a point O, the person looks up at a hot air balloon B, placed above the level of his eye. The angle which the line of sight makes with the horizontal through O is called the angle of elevation of hot air balloon B.

Real life example
Practical scenario - I

Different practical conditions are illustrated below along with relevant formulae.

(i). Say, the angle of elevation of a tower from a distance 'd' from its foot is 'θ' and height of the tower is 'h' meter(refer ΔPQR). Then
       

  • Example: A cell tower stands vertically on the ground. From a point on the ground which is 20 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 75°. Find the height of the tower ?
    Sol: Let height of the tower is BC.
    Distance between foot of the tower and the observation point is AB.
    From ΔABC, tan 75° = BC/AB
    2 + √3 = BC/20
    BC = 20(2 + √3)
    ∴ Height of the cell tower = 20(2 + √3) m.
Real life example
Practical scenario - II

(ii). Say, the angle of elevation of the top of a tower as observed from a point on the ground is θ1 and on moving 'd' meters towards the tower, the angle of elevation is θ2. Then the height of the tower h is given by
       

  • Example: The angle of elevation of the top of a tower from a certain point is 60°. If the observer moves 30 m towards the tower, the angle of elevation of the top of the tower increases by 15°. The height of the tower ?
Real life example
Practical scenario - III

(iii). Say, two men on either side of the tower and in the same straight line with its base notice the angle of elevation of top of the tower to be θ1 and θ2. If the height of the tower is 'h' meters, then the distance(AB) between the two men is given by
d = h(cot θ1 + cot θ2)        

  • Example: Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 60° and 45° respectively. If the lighthouse is 120 m high, then the distance between the two ships.
Real life example
Practical scenario - IV

(iv). Say, two poles(AB & CD) of equal height 'h' are located on either side of the road opposite to each other. Let the road(BD) be 'd' meter wide. From a point (E) between them on the road, the angle of elevations of the poles are θ1 and θ2 respectively. Then the height of the pole (h) is given by
       

  • Example: Two electric poles of equal height are standing opposite each other on either side of the road, which is 4 m wide. From a point between them on the road, the angle of elevation of the electric poles are 60° and 75° respectively. Then the height of the electric pole.
Real life example
Practical scenario - V

(v). Say, the angle of elevations of the top of a tower from the bottom and top of a building of height 'h' are θ1 and θ2 respectively. Then height of the tower(h + x) is given by =      

  • Example: The angle of elevation of the top of a tower from the bottom and top of a building of height 20 m are 75° and 30° respectively. The height of the tower ?
Real life example
Practical scenario - VI

(vi). Say, a flag staff of height 'h' is erected upon the top of a building of height 'x'. Say, at a distance 'd', the angles of elevation of the tops of the flag staff and building are θ1 and θ2 respectively. Then the height of the flag staff(h) and height of the building(x) are given by
       

  • Example: From a point 'P' on the ground the angle of elevation of a top of a 15 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flag staff from P is 60°. Find the lenght of the flag staff and the distance of the building from the point P.
Angle of Depression

An angle of depression is the angle between the horizontal and the line of sight when an observer looks downward. Suppose that from a point O, the person looks downward at a boat B, placed below the level of his eye. The angle which the line of sight makes with the horizontal through O is called the angle of depression of boat B.

Real life example
Lighthouse example

As observed from the top of a 100 meters high lighthouse from the sea-level, the angles of depression of two ships are 30° and 60°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

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