6.3 Heights and Distances

. Heights and Distances In this section, we will see how trigonometry is used for finding the heights and distances of various objects without actually measuring them. For example, the height of a tower, mountain, building or tree, distance of a ship from a light house, width of a river, etc. can be determined by using knowledge of trigonometry. The process of finding Heights and Distances is the best example of applying trigonometry in real-life situations. We would explain these applications through some examples. Before studying methods to find heights and distances, we should understand some basic definitions. Line of Sight The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. Fig. . Line of sight Trigonometry Theodolite Theodolite is an instrument which is used in measuring the angle between an object and the eye of the observer. A theodolite consists of two graduated wheels placed at right angles to each other and a telescope. The wheels are used for the measurement of horizontal and vertical angles. The angle to the desired point is measured by positioning the telescope towards that point. The angle can be read on the telescope scale. Angle of Elevation The angle of elevation is an angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level. That is, the case when we raise our head to look at the object. (see Fig. . ) Angle of Depression The angle of depression is an angle formed by the line of sight with the horizontal when the point is below the horizontal level. That is, the case when we lower our head to look at the point being viewed. (see Fig. . ) Clinometer The angle of elevation and depression are usually measured by a device called clinometer. ¾ From a given point, when height of an object increases the angle of elevation increases. If h h > then α β > ¾ The angle of elevation increases as we move towards the foot of the vertical object like tower or building. If d < then β α > Fig. . Horizontal line Angle of elevation Line of sight Object P Eye of the observer O Fig. . Angle of Depression Fig. . Fig. .