There is a bridge of the length \(h\) on a valley. The angle of depression of a temple lying in a valley from two ends of a bridge are \(\alpha\) and \(\beta\), then the height of the bridge from top of the temple \(=\) (a) \([(h \tan \alpha \tan \beta) /(\tan \alpha-\tan \beta)]\) (b) \([(h \tan \alpha \tan \beta) /(\tan \alpha+\tan \beta)]\) (c) \([(\tan \alpha \tan \beta) /\\{h(\tan \alpha-\tan \beta)\\}]\) (d) \([\\{h(\tan \alpha+\tan \beta)\\} /(\tan \alpha \tan \beta)]\)

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. It is a crucial tool in various fields such as architecture, engineering, astronomy, and even art. At its core, trigonometry provides a way to link the geometric concept of angles with the algebraic concept of functions and ratios.

Key functions in trigonometry include sine, cosine, and tangent, which are ratios derived from the sides of a right-angled triangle. These functions help solve problems that involve determining unknown lengths or angles in a triangle, such as the angle of depression problems.

A right triangle is a triangle that has one angle exactly equal to 90 degrees. The longest side, opposite to the right angle, is called the hypotenuse. The sides that form the right angle are known as the legs or catheti of the triangle.

In the context of solving real-world problems, right triangles are especially useful for modeling situations where two dimensions intersect at a right angle, such as in the case of the bridge and temple from the textbook exercise. Once a right triangle is established, trigonometric functions can be applied to find unknown measurements.

The tangent function is one of the primary trigonometric functions and is abbreviated as 'tan'. For a given angle in a right-angled triangle, the tangent is the ratio of the length of the side opposite to the angle (opposite side) to the length of the side adjacent to the angle (adjacent side). In formula terms, it is written as:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]

This function is widely used in problems involving slopes, angles of elevation, and angles of depression, which are scenarios where only a part of the triangle's sides is known, and there is a need to find the heights or distances indirectly, as demonstrated in the angle of depression problem provided.

Besides the tangent function, other fundamental trigonometric functions are sine (sin), cosine (cos), secant (sec), cosecant (csc), and cotangent (cot). Together, these functions relate the angles of a triangle to the lengths of its sides. Trigonometric functions are periodic and can be graphed on a coordinate system, which also allows their values to be determined for any given angle beyond the basic right triangle scenario.

Understanding the interrelations of these functions is essential. For example, the reciprocal of the tangent function is the cotangent function, and they can be expressed in terms of sine and cosine as: \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]. The concepts behind trigonometric functions are invaluable in solving problems such as those involving angles of depression and elevation, where the relationship between height and distance needs to be understood.