- Heights and Distances - Angle of Elevation, Angle of Depression
- Height and Distance Examples and Solutions
- line of sight, angle of elevation, angle of depression
In everyday life we come across many buildings, monuments and other structures. The heights of these structures can be found out with the help of trigonometry.
In this figure, the line AC drawn from the eye of the student to the top of the minar is called the line of sight. The student is looking at the top of the minar. The angle BAC, so formed by the line of sight with the horizontal, is called the angle of elevation of the top of the minar from the eye of the student.
Thus, the line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object.
You would need to know the following:
(i) the distance DE at which the student is standing from the foot of the minar
(ii) the angle of elevation, ∠ BAC, of the top of the minar
(iii) the height AE of the student.
Assuming that the above three conditions are known, how can we determine the height of the minar?
In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student.
To find BC, we will use trigonometric ratios of ∠ BAC or ∠ A. Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC.
Therefore, `tanA= "BC"/"AB"` or `cotA= "AB"/"BC"`, which on solving would give us BC. By adding AE to BC, you will get the height of the minar.
Now, consider the situation given below, The girl sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight with the horizontal is called the angle of depression.
Thus, the angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed
Example 1 : A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.
Solution : First let us draw a simple diagram to represent the problem (see Figure). Here AB represents the tower, CB is the distance of the point from the tower and ∠ ACB is the angle of elevation. We need to determine the height of the tower, i.e., AB. Also, ACB is a triangle, right-angled at B.
To solve the problem, we choose the trigonometric ratio tan 60° (or cot 60°), as the ratio involves AB and BC.
Now, `tan 60° = "AB"/"BC"`
`sqrt3= "AB"/ 15`
`"AB"= 15 sqrt3`
Hence, the height of the tower is`15 sqrt3` m.
Shaalaa.com | Application of Trigonometry part 1 (Concepts)
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A person standing on the bank of river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and width of the river. `(sqrt 3=1.73)`
From the top of a lighthouse, an observer looks at a ship and finds the angle of depression to be 60° . If the height of the lighthouse is 90 meters, then find how far is that ship from the lighthouse? (√3 = 1.73)
There are three stair-steps as shown in the figure below. Each stair step has width 25 cm, height 12 cm and length 50 cm. How many bricks have been used in it, if each brick is 12.5 cm x 6.25 cm x 4 cm?
Two building are in front of each other on either side of a road of width 10 metres. From the top of the first building which is 40 metres high, the angle of elevation to the top of the second is 45°. What is the height of the second building?
A person is standing at a distance of 80 m from a church looking at its top. The angle of elevation is of 45°. Find the height of the church.
From the top of a lighthouse, an observer looking at a ship makes angle of depression of 60°. If the height of the lighthouse is 90 metre, then find how far the ship is from the lighthouse.