Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

A tower \(\mathrm{AB}\) 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 \(\mathrm{C}\) at a distance \(2 \mathrm{~d}\) from \(\mathrm{C}\) 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

Option (A) is correct: \( 2 \tan \alpha = 2 \cot \beta - \cot \gamma \).

Step 1: Understand the Problem

We are given a right triangle configuration where a tower AB leans west, making an angle \( \alpha \) with the vertical. The observer at point C sees the top B at an angle \( \beta \) from a distance \( d \) directly east of A. Another observer east of C sees it at \( \gamma \) from a total distance of \( 3d \). The task is to determine the correct relationship among these angles.

Step 2: Draw the Geometry

Visualize two right triangles \( \Delta ACB \) and \( \Delta ADB \) where D is due east of C. C is at distance \( d \) from A and D is at distance \( 2d \) from C. Use the tangent of angles \( \beta \) and \( \gamma \) to express the height of the triangle.

Step 3: Use Trigonometric Identities

From triangle \( \Delta ACB \), we have \( \tan \beta = \frac{h}{d} \), where \( h \) is the height to point B.From triangle \( \Delta ADB \), where C is included, we have \( \tan \gamma = \frac{h}{3d} \).

Step 4: Express Height h

Using the equations:1. From \( \Delta ACB \): \( h = d \tan \beta \).2. From \( \Delta ADB \): \( h = 3d \tan \gamma \),Equating the two expressions for h gives: \( d \tan \beta = 3d \tan \gamma \).

Step 5: Solve for Ratios

Dividing both sides by \( d \) yields \( \tan \beta = 3 \tan \gamma \). Substitute back: \( \tan \beta - \tan \gamma = 2 \tan \gamma = 2 \tan \alpha \), leading to \( 2 \tan \alpha = 2 \cot \beta - \cot \gamma \).

Step 6: Check Against Options

Compare the derived formula to the given options. The solution matches the option (A): \( 2 \tan \alpha = 2 \cot \beta - \cot \gamma \).

Key Concepts

Angular Elevation

Angular elevation is the upward angle from the horizontal plane to a point in sight. Think of it as looking up to see the top of a tower or a hill. It helps in determining the height or slope of an object relative to the observer.

• In the problem, the angular elevation of the top of the tower (point B) is observed from two different points, each with a distinct perspective (angles \( \beta \) and \( \gamma \)).
• The angles \( \beta \) and \( \gamma \) are crucial in using trigonometric functions to build relationships with the tower’s height and its angle of lean \( \alpha \).

Using angular elevation helps connect the geometry of the situation with a few simple measurements, allowing mathematical calculations to estimate distances and heights.

Right Triangle

A right triangle is a fundamental shape in geometry characterized by one angle of exactly 90 degrees. This simple structure makes it ideal for applying trigonometric functions. In problems involving angles of elevation, right triangles are often used to solve for distances and heights.

• In our situation, the tower AB leaning towards the west forms two right triangles with the ground: \( \Delta ACB \) and \( \Delta ADB \).
• Each triangle shares a common height \( h \), which is the vertical distance from the top of the tower to the horizontal ground.

When dealing with right triangles, there's always a hypotenuse (here, the slanted position of the tower) and two legs (one representing the base on the horizontal plane and the other the height \( h \)). Understanding how to work within this triangle structure is key in tackling trigonometry problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variable. These identities allow you to simplify and transform complex trigonometric expressions into manageable forms. In our context, we focus on the tangent identity.

• The identity for tangent, \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), helps determine the height \( h \) from the angles \( \beta \) and \( \gamma \).
• Another useful identity is \( \cot \theta = \frac{1}{\tan \theta} \), which is used in rearranging terms and simplifying the expressions provided in the options.

By using these relationships, we move between tangents and cotangents depending on what is most useful to solve the geometry involved. They provide a framework for deriving connections between different parts of the problem.

Tangent Function

The tangent function is a fundamental concept in trigonometry, defined as the ratio of the opposite side to the adjacent side in a right triangle. This function is particularly useful in evaluating angles of elevation.

• In the given problem, \( \tan \beta \) and \( \tan \gamma \) are used to express the height \( h \) of the tower above the base line at two specific points.
• To solve the problem, you equate the expressions derived using \( \tan \beta \) and \( \tan \gamma \), reflecting the consistent height \( h \) across different distances.

Thus, the tangent function acts as a bridge between angles and linear dimensions, such as distance or height. Solving trigonometry problems often comes down to setting up and simplifying equations based on this relationship, as demonstrated in finding \( 2 \tan \alpha \).