Angle of Elevation Learning

Introduction:

Assume that you are walking on a road. Normally your eyes will be viewing objects at a horizontal level at a distance of your height from the ground. Suppose you see a tall building in the vicinity and you want to see the top of the building. What you do is, rotate your line of sight from horizon to spot the top of thr building. The angle by which you do that is called the Angle of Elevation.

The angle of elevation and depression are a wonderful concepts in the subject of trigonometry.


Angle of elevation and depression- Definition


Let AB be an object. The angle θ that is required to see the point A from point O is called the angle of elevation of the object at point O.

The angle of elevation of an object depends on the point from where it is measured. The more you are closer to the object, the more is the angle of elevation and vice versa.

Hence, it is a must that the angle of elevation of an object should always be accompanied by the information from where it is measured.

Angle of Depression is the angle by which you tilt your eye down from horizontal line to see an object which is below you.


Angle of elevation – Practical applications

The concept of angle of elevation and depression is used very widely in estimations of heights and depths. In fact, in trigonometry, a separate topic heights and distances deals with angle of elevation and angle of depression. The angle of depression is the angle made below the horizon for objects at lower levels.



Suppose the height of the building shown in the above picture (h) is to be estimated. It may not be practicable to measure that directly. But it is always possible to measure the angle of elevation θ (equipment for such equipment is available) and also the horizontal distance between the building and the point of angle measurement (d).

The basic trigonometric identity, tan θ =`(h)/(d)`, will easily give you the estimate.


Problems based on Angle of elevation and angle of depression


Let us do a  problems relating to angle of elevation and angle of depression to understand the concept properly.

Angle of Elevation Problems :

1)  A balloon is flying high. A boy looks at it. His eyes make an angle of 30 º from the ground to the balloon.

This is angle of elevation. The distance between the boy and  the flying balloon is 10 m. Find the height of the balloon above the Ground.

Here we use sine 30º formula =  opposite side / hypotenuse   = height / hypotenuse

Sine 30º= 1/2     hypotenuse = 10 m

so height = sine 30º x 10  =    `(1)/(2)` x 10m =  5 meter

So the ballon flies at a height of 5 meter above the ground

Problems based on Angle of depression

Problem 2)From the top of a light house, the angle of depression of 2 ships on the opposite side of the light house  are

observed as  45º and 3 angle 0º. The height of the light house is 200 m. Find the distance between the ships.


Let  M be  a ship and N be another ship.   Let AB be the light house

CD is a line drawn horizontally through B.

From B  the angle of depression is 45º and 30º.

CD is parallel to MN

So angle CBM= angle BMA and angle DBN = angle BNA.

That makes angle BMA=45º and angle BNA= 30 degrees(Interior alternate angles  are equal)

In the right triangle BAN ,  tan 30º = AB/AN

Tan 30º= `(1)/(sqrt(3))`        So `(1)/(sqrt(3))`  = `(200)/(AN)`

So AN =  200√3 = 200 x 1.732 = 346.4 m

Now let us measure AM

Tan 45º = 1   Tan 45º = `(AB)/(AM)`  so AM x 1 = AB   AM= AB   AB= 200m so AM = 200 m

Distance between Ship M and the light house is 200m

Distance between ship N and the light house is 346.4 m

Add them both.

Distance between ship M and N is 200+346.4 = 546.4 meters

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