SOME APPLICATIONS OF TRIGONOMETRY

T RIGONOMETRY 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 (see Fig. . ). Fig.

. Now, consider the situation given in Fig. . .

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 (see Fig.

Now, you may identify the lines of sight, and the angles so formed in Fig. . . Are they angles of elevation or angles of depression?

Let us refer to Fig. . again. If you want to find the height CD of the minar without actually measuring it, what information do you need?

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. In  ABC, the side BC is the opposite side in relation to the known  A. Now, which of the trigonometric ratios can we use? Which one of them has the two values that we have and the one we need to determine?

Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC. tan A = BC AB or cot A = AB, BC which on solving would give us BC. By adding AE to BC, you will get the height of the minar. Now let us explain the process, we have just discussed, by solving some problems.

Example : A tower stands vertically on the ground. From a point on the ground, which is m away from the foot of the tower, the angle of elevation of the top of the tower is found to be °. Find the height of the tower. Solution : First let us draw a simple diagram to represent the problem (see Fig.

. ). 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 ° (or cot °), as the ratio involves AB and BC. tan ° = AB BC = AB AB = Hence, the height of the tower is m. Fig.

. Example : An electrician has to repair an electric fault on a pole of height m. She needs to reach a point .3m below the top of the pole to undertake the repair work (see Fig. .

). What should be the length of the ladder that she should use which, when inclined at an angle of ° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take = .

) Solution : In Fig. . , the electrician is required to reach the point B on the pole AD. BD = AD – AB = ( – .

)m = . m. Here, BC represents the ladder. We need to find its length, i.e., the hypotenuse of the right triangle BDC.

Now, can you think which trigonometic ratio should we consider? It should be sin °. BD BC = sin ° or . BC = BC = .

 = . m (approx.) i.e., the length of the ladder should be . m. DC BD = cot ° = DC = .

= . m (approx.) Therefore, she should place the foot of the ladder at a distance of . m from the pole. Fig.

. Example : An observer . m tall is . m away from a chimney.

The angle of elevation of the top of the chimney from her eyes is °. What is the height of the chimney? Solution : Here, AB is the chimney, CD the observer and  ADE the angle of elevation (see Fig. .

). In this case, ADE is a triangle, right-angled at E and we are required to find the height of the chimney. We have AB = AE + BE = AE + . and DE = CB = .

m To determine AE, we choose a trigonometric ratio, which involves both AE and DE. Let us choose the tangent of the angle of elevation. tan ° = AE DE = AE . AE = .

So the height of the chimney (AB) = ( . + . ) m = m. Example : From a point P on the ground the angle of elevation of the top of a m tall building is °.

A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is °. Find the length of the flagstaff and the distance of the building from the point P. (You may take = . ) Solution : In Fig.

. , AB denotes the height of the building, BD the flagstaff and P the given point. Note that there are two right triangles PAB and PAD. We are required to find the length of the flagstaff, i.e., DB and the distance of the building from the point P, i.e., PA.

Fig. . Since, we know the height of the building AB, we will first consider the right  PAB. We have tan ° = AB AP = AP AP = i.e., the distance of the building from P is m = .

m. Next, let us suppose DB = x m. Then AD = ( + x ) m. Now, in right  PAD, tan ° = AD AP  =