The angle of depression for two consecutive km stones on a horizontal road observed on the opposite sides of plane from a plane are \(\alpha\) and \(\beta\) respectively and if the height of the plane is \(h\) then \(h=\) (a) \([(\tan \alpha-\tan \beta) /(\tan \alpha \tan \beta)]\) (b) \([(\tan \alpha \tan \beta) /(\tan \alpha-\tan \beta)]\) (c) \([(\tan \alpha+\tan \beta) /(\tan \alpha \tan \beta)]\) (d) \([(\tan \alpha \tan \beta) /(\tan \alpha+\tan \beta)]\)

The concept of the tangent of an angle plays a vital role in solving problems related to angles of elevation and depression. When we are dealing with right triangles, the tangent of an acute angle is the ratio of the opposite side to the adjacent side. Mathematically, for a right triangle with an acute angle designated as \( \theta \), this ratio is represented as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

This trigonometric ratio is particularly useful when we have to find the height of an object, such as a plane in flight, or the distance from a point of observation, such as the distance of a plane from a ground marker. Understanding how to manipulate the tangent function is essential for solving many real-world problems involving right triangles.

Trigonometry is widely used when it comes to understanding and calculating the dimensions within right-angled triangles. In the context of the given problem, we deal with trigonometry by working out the height of a plane based on the angles of depression. The key to solving such problems is to draw a clear diagram and label all known and unknown variables. The diagram reduces the complexity of the problem and provides a visual representation, which makes it easier to relate the trigonometric functions to the triangle's sides.

The tangent ratios derived from the angles of depression give us equations that are essential in representing the sides of the triangles in terms of the height of the plane. This process allows us to deal with unknown distances and is an excellent example of the practical applications of trigonometry in right triangles.

Trigonometric ratios are crucial tools in the toolkit of anyone dealing with trigonometry. These ratios—sine, cosine, and tangent—are relationships that exist between the angles and sides of right-angled triangles. In our specific angle of depression problem, we use the tangent ratio to connect the angles \( \alpha \) and \( \beta \) to the height of the plane, \( h \).

When we derive our two key equations using the tangent of the angles \( \tan \alpha = h/x \) and \( \tan \beta = h/y \), these reflect the very essence of trigonometric ratios in action. Furthermore, familiarizing oneself with the manipulation of these ratios is essential for solving and understanding the step-by-step solutions. Such manipulation involves finding a common denominator, adding and subtracting expressions, and solving for the unknown, as demonstrated in our exercise. Trigonometric ratios are indispensable in not only mathematics but also in various fields such as engineering, physics, and even geography.