Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Due south of a tower which is leaning towards north there are two stations at distances \(\mathrm{x}\) and y respectively from its foot. If \(\alpha, \beta\) respectively be the angles of elevation of the top of the tower at these stations, then the inclination \(\theta\) of the tower to the horizontal is given by \(\cot \theta=\) (A) \(\frac{y \cot \alpha-x \cot \beta}{y-x}\) (B) \(\frac{y \cot \alpha+x \cot \beta}{y-x}\) (C) \(\frac{y \cot \alpha-x \cot \beta}{y+x}\) (D) \(\frac{y \tan \alpha-x \sin \beta}{y-x}\)

Step-by-Step Solution

Answer

Option (A): \(\cot \theta = \frac{y \cot \alpha - x \cot \beta}{y-x}\).

Step 1: Understand the Problem

We have a leaning tower, which is inclined from the vertical, and two observation stations located south of the tower at distances \(x\) and \(y\). The angles of elevation from these stations to the top of the tower are \(\alpha\) and \(\beta\), respectively. We need to find an expression for the inclination \(\theta\) of the tower to the horizontal.

Step 2: Trigonometric Relationships

At each station, we apply the trigonometric relationship of the cotangent for the angles of elevation. The formula for cotangent in terms of height \(h\) of the tower and the horizontal distance (distance of the station from the foot of the tower projected along the ground) is: \(\cot \alpha = \frac{h}{x'}\) where \(x'\) accounts for the horizontal offset due to leaning.

Step 3: Apply to Both Stations

Similarly, for station at distance \(y\), we have: \(\cot \beta = \frac{h}{y'}\) where \(y'\) is the horizontal distance after correction for the tower's incline. Note that due to the identical nature of the problem setup, \(x'\) and \(y'\) can be related to the angle \(\theta\).

Step 4: Develop Relationship Between \(\theta\), \(\alpha\), and \(\beta\)

The inclination \(\theta\) plays a role in modifying the effective base distances. Using similar triangles and relationships, we equate them to form an equation \(\tan \theta = \frac{x - x'}{h}\) and \(\tan \theta = \frac{y - y'}{h}\). Rearranging, we use cotangent identities to form expressions dependent on \(\alpha\) and \(\beta\).

Step 5: Solve for \(\cot \theta\)

Solving these rearranged equations and combining them, we find: \(\cot \theta =\frac{y \cot \alpha - x \cot \beta}{y-x}\) which matches option (A) in the problem statement.

Key Concepts

Angles of Elevation

In trigonometry, the angle of elevation is the angle between the horizontal plane and the line of sight pointing upwards to an object. When you stand at a certain point and look up at the top of a tower or a building, that's the angle of elevation.
Imagine you're at an observation station looking up at the top of a leaning tower. From your viewpoint, the angle your line of sight makes with the horizontal is what we call the angle of elevation. This angle is crucial because it helps us connect the height and distance through trigonometric ratios. Differences in these angles from different positions give us information about the inclination or tilt of the structure.
Calculating this angle correctly can be crucial for determining the correct angles and thus understanding the structure we're analyzing.

Cotangent Identities

The cotangent is a fundamental trigonometric function. It is defined as the reciprocal of the tangent, or \( \cot \theta = \frac{1}{\tan \theta} \), which also equals the ratio of the adjacent side to the opposite side in a right triangle.
This identity is particularly helpful when working with inclined or leaning structures. It simplifies comparisons between different angles of elevation from multiple locations.
Using cotangent identities, we can express complicated trigonometric relationships in simpler terms. For instance, in the case of our leaning tower problem, we can leverage \( \cot \alpha = \frac{h}{x'} \) and \( \cot \beta = \frac{h}{y'} \) to derive expressions that help determine the inclination angle \( \theta \). This identity offers a powerful tool to work with multiple observations from different viewing points and reduce them down to a desired relationship.

Inclination of a Tower

When we speak of the inclination of a tower, we're referring to how far the tower deviates from being perfectly vertical. In mathematical terms, this inclination is represented using theta (\( \theta \)).
For a leaning tower or any such structure, this angle is significant, as it indicates the degree to which the tower leans from the vertical. In our problem, this leaning is observed from two stations, and both their observations must be combined to determine \( \theta \).
Using trigonometric relationships and identities, the inclination can be calculated even without direct measurement. In the problem solution, we specifically find the cotangent of the angle \( \theta \) which represents the inclination based on observations of angles of elevation \( \alpha \) and \( \beta \). Successfully solving for \( \cot \theta \) helps us understand how the tower’s tilt is related to these observed angles.

Trigonometric Relationships

Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles.
In the problem we’re working with, these trigonometric relationships help us tie together various angles, distances, and heights involved.
For the problem at hand, these relationships allow us to connect the concept of elevation with inclination. Using basic trigonometric identities and laws such as those involving tangent and cotangent, we're able to express one part of the problem in terms of another.
For example, by relating angles \( \alpha \) and \( \beta \) to \( \theta \) through cotangent identities, we can form essential equations. These equations are then solved to find the tower's inclination.

• Understanding these relationships provides a mathematical map to navigate through complex geometric scenarios.
• The correct application of trigonometric concepts is key to accurately determining real-world features like the leaning of a tower.