The angle of depression of the top and bottom of a tower observed from top of a lighthouse of 300 meter height are \(30^{\circ}\) and \(60^{\circ}\) respectively then the height of the tower is (a) 300 meter (b) \(100 \mathrm{~m}\) (c) \(200 \mathrm{~m}\) (d) \(50 \mathrm{~m}\)

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right-angled triangles.

The basic trigonometric functions are sine, cosine, and tangent, which are ratios of the sides of a right triangle. In such triangles, one angle is always 90 degrees, and the remaining angles must sum to 90 degrees.

When dealing with the angle of depression problems, we're often looking at two right triangles that share a common side – the height from the observer's eye level, such as the height of a lighthouse in the given problem. To solve these problems, we use trigonometric ratios to relate angles to the lengths of sides.

Identifying Right Triangles

• Firstly, conceptualize the problem by sketching a diagram with all given angles and measurements. The angle of depression is equal to the angle of elevation from the lower point to the observer's eye level.
• Secondly, identify the right triangles in the situation. In our problem, two right triangles are formed by the observer's eye line from the lighthouse and the lines of sight to the top and bottom of the tower.

Understanding these fundamental concepts allows students to solve a wide range of real-world problems, making trigonometry a powerful tool in various fields.

In trigonometry, the tangent ratio is particularly useful when dealing with right triangles, as it compares the length of the side opposite to an acute angle (the angle less than 90 degrees) with the length of the side adjacent to that angle. Mathematically, it is expressed as: \begin{center}\[\text{tan(angle)} = \frac{\text{opposite}}{\text{adjacent}}\]For the angle of depression problem, we apply this formula twice for the two angles of depression given, with the height of the lighthouse as the 'opposite' side and the distances to the top and bottom of the tower playing the roles of 'adjacent' sides.

Applying the Tangent Ratio

• To find the distance to the top of the tower (DT), we use the tangent of angle A1 and the known height of the lighthouse (HL).
• For the distance to the bottom (DB), we use the tangent of angle A2 and the same height (HL).

This approach helps in solving the two separate equations to find DT and DB, which are crucial in calculating the height of the tower.

Height and distance problems are a common application of trigonometry, often requiring the measurement of an object's height without the need for physical measurement. The problem we're given is an example of how these principles are applied.

By observing an object from a distance and knowing the height of the vantage point and the angles of depression, you can calculate the height of the object using trigonometric functions.

Solving Height and Distance Problems

• Begin by charting out a representation, incorporating all given heights and angles.
• Identify the relevant sides and angles for each right triangle formed in the context.
• Use trigonometric functions, like the tangent ratio, to establish equations that can be solved for the unknown distances to the object.
• In the final step, apply arithmetic operations to deduce the height or distance you're tasked with finding.

These methods are vital in fields ranging from surveying and navigation to engineering and astronomy. It's essential to ensure the accuracy of all known values and measured angles to arrive at a reliable solution.