Problem 5 - A Complete Resource Book in Mathematics for JEE Main

Question

An observer finds that the angular elevation of a tower is \(A\). On advancing \(3 \mathrm{~m}\) towards the tower the elevation is \(45^{\circ}\) and on advancing \(2 \mathrm{~m}\) nearer, the elevation is \(90^{\circ}\) \(-A\). The height of the tower is (A) \(2 \mathrm{~m}\) (B) \(4 \mathrm{~m}\) (C) \(6 \mathrm{~m}\) (D) \(8 \mathrm{~m}\)

Step-by-Step Solution

Answer

The height of the tower is 6 meters.

Step 1: Understand the Problem

The problem involves an observer moving towards a tower and noting the angles of elevation at different positions. We need to find the height of the tower using these angular measurements.

Step 2: Set Up the Geometry

The observer initially observes an angle of elevation \( A \). After moving 3 meters closer, the angle is \( 45^\circ \). Then, moving 2 more meters closer, the angle becomes \( 90^\circ - A \). We can represent the initial distance as \( x \) meters.

Step 3: Apply Trigonometric Function for First Position

From initial position with angle \( A \), the height \( h \) can be represented as \( h = x \tan A \).

Step 4: Apply Trigonometric Function for Second Position

After moving 3 meters (total distance \( x - 3 \)), the angle is \( 45^\circ \). For \( \tan 45^\circ = 1 \), this gives \( h = (x - 3) \cdot 1 = x - 3 \).

Step 5: Apply Trigonometric Function for Third Position

After moving 5 meters (total distance \( x - 5 \)), the angle is \( 90^\circ - A \). Using the property \( \tan (90^\circ - A) = \cot A \), the height is \( h = (x - 5) \cot A \).

Step 6: Create Equations from Trigonometric Relationships

1. From Step 3: \( h = x \tan A \),2. From Step 4: \( h = x - 3 \),3. From Step 5: \( h = (x - 5) \cot A \).

Step 7: Simplify and Solve Equations

Equating equations from Steps 3 and 4: \[ x \tan A = x - 3 \quad \Rightarrow \quad x(\tan A - 1) = -3 \]Equating equations from Steps 3 and 5: \[ x \tan A = (x - 5) \cot A \quad \Rightarrow \quad x \tan A = \left(\frac{x}{\tan A} - \frac{5}{\tan A}\right) \]Solve these to find expressions for height \( h \).

Step 8: Calculate the Height of the Tower

From equated expressions, \( h = 6 \) m. Substitute back to confirm consistency in all equations. Therefore, height of the tower is 6 m.

Key Concepts

Angle of Elevation

The angle of elevation is a critical concept in trigonometry. It refers to the angle formed between the horizontal plane and the line of sight when an observer looks upwards to view an object, like a tower or a tall building. These angles are crucial in various real-life applications, including navigation, architecture, and even in everyday tasks.
For example, in this exercise, the observer measures the angle of elevation at different distances from the tower to determine its height. The observer moves toward the tower and records the angles as they change, which are key in solving for the tower's height.
Understanding angles of elevation helps in effective problem solving. By recognizing how these angles serve as visual cues, one can determine unknown heights or distances using trigonometric functions like tangent or cotangent.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions that are valid for every angle. They play a significant role in solving problems relating to angles and heights, just like in this exercise.
For instance, in the problem, we employ identities such as \( \tan(90^\circ - A) = \cot A \). This particular identity is used to transform the expression for the angle in the third position from tangent to cotangent, aiding in the simplification and solution of equations.
Recognizing and using these identities can greatly simplify complex calculations. They allow us to express one trigonometric function in terms of another, making it easier to solve problems when dealing with angles. The use of identities is crucial for efficient and effective problem-solving in trigonometry.

Problem-Solving Strategy

Effective problem-solving in trigonometry often involves a step-by-step strategy to ensure all elements of a problem are addressed. In this exercise, the strategy begins by understanding the problem and setting up the geometric scenario. Observational data, such as angles of elevation at various positions, is critical to defining the problem clearly.
Next, applying the appropriate trigonometric functions to these observations, as seen here, involves determining expressions for the height of the tower using the angles measured. Equations are formed from these trigonometric relationships, like \( h = x \tan A \) and \( h = (x - 5) \cot A \), among others.
The final step involves simplifying and solving these equations systematically to find the desired quantity, the height of the tower in this case. This involves equating and rearranging the expressions until a consistent solution is found, which in this exercise is 6 meters.
By breaking the problem down into manageable steps, and by checking calculations at each stage, one can ensure accuracy and efficacy in solving trigonometric problems.