Question

The angle of elevation of the top of a tower standing on a horizontal plane from a point \(A\) is \(\alpha\). After walking a distance \(d\) towards the foot of the tower the angle of elevation is found to be \(\beta\). The height of the tower is- (1) \(\frac{d}{\cot \alpha+\cot \beta}\) (2) \(\frac{d}{\cot \alpha-\cot \beta}\) (3) \(\frac{d}{\tan \beta-\tan \alpha}\) (4) \(\frac{d}{\tan \beta+\tan \alpha}\)

Short answer

3. \[ \frac{d}{\tan \beta - \tan \alpha} \]

Step-by-step solution

Understand the Problem

Given a point A from which the angle of elevation to the top of a tower is \(\alpha\). After walking a distance \(d\) towards the tower, the angle of elevation changes to \(\beta\). The problem is to find the height of the tower.

Define Variables

Let's denote the height of the tower as \(h\). The initial distance from point A to the base of the tower is \(x\).

Set up the First Equation Using Trigonometry

From point A, \[ \tan(\alpha) = \frac{h}{x} \]

Set up the Second Equation After Walking Distance

After moving a distance \(d\) towards the tower, the new distance to the base is \(x - d\). The angle of elevation is \(\beta\). Thus, \[ \tan(\beta) = \frac{h}{x-d} \]

Express x in Terms of h and α

From \[ \tan(\alpha) = \frac{h}{x} \], rearranging gives, \[ x = \frac{h}{\tan(\alpha)} \]

Substitute x in the Second Equation

Substitute \[ x = \frac{h}{\tan(\alpha)} \] into the second equation: \[ \tan(\beta) = \frac{h}{\frac{h}{\tan(\alpha)} - d} \]

Simplify the Equation

Simplify the fraction: \[ \tan(\beta) = \frac{h}{\frac{h - d\tan(\alpha)}{\tan(\alpha)}} \] \[ \tan(\beta) \cdot \frac{h - d \tan(\alpha)}{\tan(\alpha)} = h \]

Solve for h

Rearrange and solve for \ h \ by isolating it on one side of the equation: \[ h( \tan(\beta) - \tan(\alpha) ) = d \tan(\alpha) \tan(\beta) \] \[ h = \frac{d \tan(\alpha) \tan(\beta)}{ \tan(\beta) - \tan(\alpha) } \]

Match the Answer

Compare the final expression with the given options. The correct option is \[ \frac{d}{\tan \beta - \tan \alpha} \] which matches option 3.

Key concepts

Understanding the Angle of Elevation

When you look up at an object above you, like the top of a tower, the angle between the line of sight and the horizontal ground is the **angle of elevation**. This concept is key in many trigonometry problems, especially in IIT JEE exams. In the context of our exercise, the angle of elevation changes as you move closer to the tower. Initially, from point A, you have one angle (\(\alpha\)). After walking closer by a distance \(d\), you get another angle (\(\beta\)). Understanding how these angles work helps in setting up the correct trigonometric equations to solve for the height of the tower.

Utilizing Trigonometric Identities

Trigonometry relies heavily on trigonometric identities, which are equations involving trigonometric functions. For this problem, the main functions are tangent (\(\tan\)). By definition, for an angle in a right triangle, \(\tan(\theta) = \frac{opposite}{adjacent}\). This identity helps us relate the tower's height to the distances involved. In the problem:

• From point A, \(\tan(\alpha) = \frac{h}{x}\)
• After walking a distance \(d\), \(\tan(\beta) = \frac{h}{x-d}\)

Using these identities, we can form equations and substitute variables to eventually isolate and solve for the unknown height \(h\).

Step-by-Step Problem-Solving

Solving trigonometry problems involves following logical and systematic steps. Let's break down the steps for our problem:

• Step 1: Understand the Problem - Identify what you need to find (height \(h\)) and what's given (angles \(\alpha\) and \(\beta\), distance \(d\)).
• Step 2: Define Variables - Assign symbols to unknowns (height \(h\), initial distance \(x\)).
• Step 3: Setup Trigonometric Equations - For point A, \(\tan(\alpha) = \frac{h}{x}\); after walking, \(\tan(\beta) = \frac{h}{x-d}\).
• Step 4: Isolate Variables - From \(\tan(\alpha)\), get \(x = \frac{h}{\tan(\alpha)}\).
• Step 5: Substitute and Simplify - Plug \(x\) in the second equation and simplify to express \(h\).
• Step 6: Solve - Solve the equation \(h ( \tan(\beta) - \tan(\alpha) ) = d \tan(\alpha) \tan(\beta)\) to get \(h\).

Following these structured steps not only helps in solving this problem but can be applied to other trigonometric problems too. Practice these steps to master trigonometric problem-solving.