Heights and Distances

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One of the main use of trigonometry is to find the distances between points, or the heights of objects, without actually measuring these distances or these heights.

Angle of elevation

Let O = Position of the observer

P = Position of the object

OH = Horizontal line through O (line of reference)

Angle of depression

Let O = Position of the observer

P = Position of the object

OH = Horizontal line through O (line of reference)

Example:

A person standing on the bank of a river, observes that the angle subtended by a tree on the bank is 60^o. When he retreats 20m from the bank, he finds the angle to be 30^o. Find the height of the tree and the breadth of the river.

Suggested answer:

In the adjoining figure,

AB = width of river = x

BC = height of tree = h

OA = 20m

O is the position of the person after retreating 20m from the bank.

From D OBC, we have

From D ABC, we have

From (i) and (ii), we get

= 17.32m