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High School Mathematics - 2
10.2 Heights and Distances

One of the most important applications of trigonometry is the application of measurement of heights and distances which cannot be measured directly.
It is also extensively used by astronomers in determining the distance of the heavenly bodies like the sun and moon and stars.

Two angles are very often used in the practical applications of trigonometry .
Line of sight
The line of object is the line from our eyes to object.

Angle of Elevation
If the object is above the horizontal level of our eyes, we have to turn our head upwards to view the object. This is called the angle of elevation.

Angle of Depression
If the object is below the horizontal level of our eyes, we have to bend our head to view it. This is called the angle of elevation.

Example 1: The angle of elevation of the top of a tower at a distance of 100 metres from its foot on a horizontal plane is found to be 60o. Find the height of the tower.
Let CA be the tower equal to h metres in length and B point at a distance of 100 metres from its foot C. It is given that angle ABC = 60o

From the right angle triangle ABC we have
h/100 = tan 60
Solving the above we get h = 173.2 metres
Answer: Height of the tower is 173.2 metres

Example 2: From the top of a cliff, 200 metres high, the angle of depression of the top and bottom of a tower are observed to be observed to be 30o and 60o, find the height of the tower.

Solution: Let AB represent the tower and P the top of cliff MP. If PX be the horizontal line through P, then angle XPA = 30o and angle XPB = 60o. Let the height of the tower be h metres. From A draw AL perpendicular to PM.
ML = AB = h=> LP = (200-h)
Again, ang PBM = ang XPB = 60
ang PAL = ang XPA = 30 (alternate angles )
From the right triangle PMB, BM/200 = cot 60
BM = 200 cot 60 = 200/3
From the right triangle PLA, AL /AP = cot 30 =>AL = LP cot 30 = =(200-h)3
But AL = BM = =(200-h)3 =200/3
Answer: h = 133 1/3 metres

Directions: Answer the following questions. Also write at least 10 examples of your own.
Q 1: The horizontal distance between two towers is 60 metres and the angular depression of the top of the first as seen from the top of the second, which is 150 metres is 30o. Find the height of the first.
100 metres
120 metres
115.36 metres

Question 2: This question is available to subscribers only!

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