CBSE Class 10 Mathematics Some Applications of Trigonometry Notes Set 03

Heights and Distances

The line of sight is the line from the observer's eye to the object. The angle of elevation is measured upward from the horizontal through the eye; the angle of depression is measured downward from it. By alternate angles, the angle of depression of an object seen from a height equals the angle of elevation of the observer measured from the object.
Always draw a figure with a horizontal line through the eye, mark the heights and distances, and look for the right triangles. Use tan = height / distance, sin = height / line of sight and cos = distance / line of sight.

Typical Problem Types

One angle: height = distance x tan(angle). Two angles with a movement: write the height from each position, equate them and solve; for movement d towards the foot, \( h=\frac{d\tan x\tan y}{\tan y-\tan x} \). A tower with a flagstaff on top: use the angle to the bottom and to the top from the same point. Two points on opposite sides of a building: the distance between them is \( h(\cot\alpha+\cot\beta) \); on the same side it is \( h(\cot\beta-\cot\alpha) \), with the nearer point having the larger elevation angle.
Complementary elevations at distances p and q from the foot give height \( \sqrt{pq} \). Speed problems: convert distance covered to a rate; 720 km/h equals 200 m/s. Boat approaching a cliff: the boat covers the difference of the two horizontal distances in the given time, and the remaining distance takes proportionally less time.