Problem 31 - A Complete Resource Book in Mathematics for JEE Main

Question

A tower \(A B\) leans towards west making an angle \(\alpha\) with the vertical. The angular elevation of \(B\), the top most point of the tower is \(\beta\), as observed from a point \(C\) due east of \(A\) at a distnace \(d\) from \(A\). If the angular elevation of \(B\) from a point due east of \(C\) at a distance \(2 d\) from \(C\) is \(\gamma\), then (A) \(2 \tan \alpha=2 \cot \beta-\cot \gamma\) (B) \(2 \tan \alpha=3 \cot \beta-\cot \gamma\) (C) \(\tan \alpha=\cot \beta-\cot \gamma\) (D) none of these

Step-by-Step Solution

Answer

The answer is (C) \( \tan \alpha = \cot \beta - \cot \gamma \).

Step 1: Understand the Problem

We have a leaning tower \(AB\) making an angle \(\alpha\) with the vertical towards the west. A point \(C\) is due east of \(A\) at a distance \(d\), and the angle of elevation to the top \(B\) from \(C\) is \(\beta\). Moving further east to \(2d\) from \(C\), the angle of elevation to \(B\) is \(\gamma\). We need to find a relation between \(\alpha\), \(\beta\), and \(\gamma\).

Step 2: Establish Trigonometric Relationships

Using trigonometry, \( \tan \beta = \frac{h}{d} \) and \( \tan \gamma = \frac{h}{3d} \), where \(h\) is the height of the tower above \(A\). Also, due to the leaning angle \(\alpha\), the horizontal shift of \(B\) can be considered as \(h \cdot \tan \alpha\).

Step 3: Set Up Equations for Heights and Positions

Since \( \beta \) and \( \gamma \) are angles of elevation from different positions, the horizontal offset due to the leaning must account for this. Thus, \( h \tan \alpha = d (\cot \beta - \cot \gamma) \).

Step 4: Analyze Each Option

Comparing the derived equation \( h \tan \alpha = d (\cot \beta - \cot \gamma) \) with the given options:- (A) does not match since factor mismatch- (B) mismatches due to factor 2 and 3- (C) matches our derived equation formThus, \( \tan \alpha = \cot \beta - \cot \gamma \).

Step 5: Verify Correctness of Final Equation

Ensure all angles and assumptions align with the geometric relations formed between the tower's positions. The derived relation correctly decomposes the height and angle factors.

Key Concepts

Angle of Elevation

When we talk about the angle of elevation in trigonometry, we refer to the angle formed between the horizontal and a line that goes from the observer's eye to an object that is located above the horizontal level. This concept is crucial in the Leaning Tower problem as it helps in determining the height of the tower from different observation points. To calculate the angle of elevation, you might use the trigonometric function \( \tan \beta = \frac{h}{d} \), where \( \beta \) is the angle of elevation, \( h \) is the height of the object, and \( d \) is the distance from the observer to the object. Understanding this helps in figuring out how different positions affect your ability to measure angles and distances in trigonometric problems. It is important to note that angles change based on the observer's position, which is precisely utilized in solving the problem related to the leaning tower.

Leaning Tower Problem

The Leaning Tower problem typically examines scenarios where a structure is not perfectly vertical. Instead, it leans at an angle. In this problem, the tower leans at angle \( \alpha \) relative to the vertical plane. The challenge lies in combining this angle with other observed angles of elevation to find a relationship between them.To calculate the effect of the leaning, adjustments to measurements are needed. For instance:

• The tower's height apparent from a point can be affected by its lean.
• The horizontal shift due to the angle can be expressed as \( h \cdot \tan \alpha \).

This shift is key in figuring out the relationship between different angles as one moves around the tower. Solving such problems requires a good understanding of how to use trigonometric identities and adjustments in these contexts.

Trigonometric Identities

Trigonometric identities play a vital role in solving complex mathematical problems as in the Leaning Tower problem. These identities help relate different angles and lengths through mathematical expressions.For the given problem, the identities \( \tan \beta = \frac{h}{d} \) and \( \tan \gamma = \frac{h}{3d} \) come into play. Additionally, it's essential to know the identity that relates angles of elevation from observer points at different distances.A crucial identity used here is:

• \( \cot x = \frac{1}{\tan x} \)

This identity helps to express equations in a form that allows for simplification and easier comparison.Understanding these identities allows you to not only solve the problem at hand but also apply these principles to variants of this type of trigonometry challenges.