Problem 58 - A Complete Resource Book in Mathematics for JEE Main

Question

From the top of a building of height \(\mathrm{h}\), a tower standing on the ground is observed to make an angle \(\theta .\) If the horizontal distance between the building and the tower is \(\mathrm{h}\), then height of the tower is (A) \(\frac{2 h \sin \theta}{\sin \theta+\cos \theta}\) (B) \(\frac{2 h \tan \theta}{1+\tan \theta}\) (C) \(\frac{2 h}{1+\cot \theta}\) (D) \(\frac{2 h \cos \theta}{\sin \theta+\cos \theta}\)

Step-by-Step Solution

Answer

Option (B): \( \frac{2 h \tan \theta}{1+\tan \theta} \) is correct.

Step 1: Understand the Problem

We have a building and a tower, with the height of the building being \( h \), and the horizontal distance between them also being \( h \). We need to find the height of the tower given the angle \( \theta \) between them.

Step 2: Create a Right Triangle

Visualize or draw a right triangle where one vertex is at the top of the building, another at the base of the tower, and the third at the point where a horizontal line from the top of the building meets the line to the top of the tower.

Step 3: Identify Triangle Components

In the right triangle, the horizontal leg is \( h \), the angle opposite the height of the tower is \( \theta \), and the vertical leg represents the height of the tower minus the height \( h \) of the building.

Step 4: Apply Trigonometric Ratio

Using the tangent function for the angle \( \theta \), set up the equation involving the opposite and adjacent sides: \( \tan \theta = \frac{\text{height of tower} - h}{h} \).

Step 5: Solve for Tower Height

Rearrange the equation \( \tan \theta = \frac{\text{height of tower} - h}{h} \) to solve for the height of the tower: \( \text{height of tower} = h \tan \theta + h \).

Step 6: Substitute Variables

Substitute the variables to match one of the given options: \( \text{height of tower} = h (1 + \tan \theta) \). Simplify to see if it matches one of the options.

Step 7: Select the Correct Answer

Compare the derived expression with the given options and identify which one matches. Notice that the expression \( h (1 + \tan \theta) \) can be rewritten to simplify further or check against each option to see if a match is found.

Key Concepts

Right Triangle

In trigonometry, the right triangle is a fundamental concept crucial for various calculations, especially in problems like the one presented. A right triangle is a three-sided polygon where one angle is exactly 90 degrees, known as the right angle. The two sides forming the right angle are called the legs, while the side opposite the right angle is known as the hypotenuse.

Understanding right triangles is essential for solving problems involving distances and heights, such as the one between the building and the tower. Here, you can think of the triangle as having:

• One leg representing the horizontal distance \( h \) between the building and the tower.
• The other leg being equivalent to the difference in height between the top of the building and the top of the tower.
• The angle at the building top where the observation is made represents the angle of elevation. This angle helps us find unknown lengths using trigonometric functions.

You create a right triangle whenever you have a perpendicular relation, especially when looking for unknown heights or distances.

Trigonometric Ratios

Trigonometric ratios are derived from the relationships between the angles and sides of right triangles. They are essential in solving problems where some dimensions are unknown, like finding the height of a tower observed from a distance. The primary trigonometric ratios are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).

In our exercise:

• \( \tan \theta \) is the ratio we used, which relates the opposite side to the adjacent side. For the problem, it is written as \( \tan \theta = \frac{\text{height of tower} - h}{h} \).

The tangent function is particularly useful here because it involves the vertical and horizontal legs of the triangle, directly applicable to determining unknown heights or distances.
By rearranging the formula, we solve for the height of the tower:

• Height of the tower = \( h \tan \theta + h \).

Using trigonometric ratios allows us to apply mathematical relationships and derive unknown values, making them incredibly powerful for geometry and trigonometry problems.

Angle of Elevation

The angle of elevation is a crucial concept in trigonometry, particularly when observing objects from a lower point to a higher point, like from the top of a building to the peak of a tower. This angle is found between the horizontal sight line and the line of sight up to the object.

In scenarios like our problem:

• The angle of elevation is marked as \( \theta \) from the observer's point at the top of the building to the top of the tower.
• It's a vital measure, because it allows us to apply trigonometric ratios like tangent to calculate the height of the tower.

The angle of elevation helps to tie together various aspects of the right triangle. Integrating it effectively with horizontal distances and trigonometric ratios allows problem solvers to figure out unknown dimensions succinctly and accurately.
Understanding and using the angle of elevation can greatly simplify problems involving multiple heights and angles, providing a clear pathway to find unknown values using the known dimensions and angles.