Sometimes , we need to find the Height of tower , building , tree , distance of a ship from light house, width of river and Angle subtended by any object at a given point, etc.

We cannot measure them accurately , though we can find them using the concepts of ,

(1) Angle of elevation

(2) Angle of depression

Angle of elevation is defined as an angle subtended by an eye with the horizontal axis to see an object in the upward direction.

Let P be the position of the object above the horizontal line OA and O be the eye of the observer, then     ∠AOP is called angle of elevation.

Here, the line joining the eye to the object  i.e. OP is called the line of sight.

* If the angle of observer moves towards the perpendicular line(tower/building) , then the angle of elevation increases and if the observer moves away from the perpendicular line (tower/building) , then the angle of elevation decreases.

* If the angle of elevation of Sun above a tower decreases then the length of shadow of a tower increases.

* If the height of tower is doubled and the distance between the observer and foot of the tower is also doubled, then the angle of elevation remains the same.

Example ,

The string of a kite is 150m long and it makes an angle of 60 degree with the horizontal . The height of the kite from the ground is what ?

Solution,

Let 'h' be the height of the kite from the ground and AB be the length of string = 150m

In right angled triangle ABC,

AC =  sin 60

AB

=>  h        =   √3 

      150           2

=>  h = 75 √3m

Hence, the height of kite from the ground is 75 √3m.

Angle of depression is defined as an angle subtended by an eye with the horizontal axis to see an object in the downward direction.

Let P be the position of the object below the horizontal line OA and O be the eye of the observer, then  ∠AOP is called angle of depression.

In case of angle of depression , the horizontal line through observer must be represented by dotted line.

          * The angle of elevation a point P as seen from a point O is always equal to the angle of depression of O as seen from P.

          * Angle of elevation and depression are always acute angle .

1.  If 'b' is height of building and the observer is x units away from base, then,

2. If 'DC=b' be the height of a building and Θ1 and Θ2 are elevations measured at B and A, respectively along the same straight line which are x units apart then,

Example ,

The angle of depression of two ship from the top of a light house are 45 deg and 30 deg towards East. If the ships are 200m apart , the height of the light house is what?

Solution,

Here applying shortcut method ,

given, x = 200m , Θ1 = 45 deg,  Θ2 = 30 deg

h =          x           

      cot Θ2 - cot Θ1

putting in the values , we have,

h =   200   m

       √3 - 1 

Here are some more few exercises for you to solve ,