Problem 37 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If from the top of a tower, 60 metre high, the angles of depression of the top and floor of a house are \(\alpha\) and \(\beta\) respectively and if the height of the house is \(\frac{60 \sin (\beta-\alpha)}{x}\), then \(\mathrm{x}=\) (A) \(\sin \alpha \sin \beta\) (B) \(\cos \alpha \cos \beta\) (C) \(\sin \alpha \cos \beta\) (D) \(\cos \alpha \sin \beta\)

Step-by-Step Solution

Answer

The answer is (B) \( x = \cos \alpha \cos \beta \).

Step 1: Understanding the Problem

We need to determine the value of \( x \) given the angles of depression from a tower to the top and floor of a house, and the expression for the house's height in terms of trigonometric functions.

Step 2: Setting Up the Scenario

Visualize the problem by imagining a triangle formed by the line of sight from the top of the tower to the top of the house. The base of this triangle is horizontal, and the line of sight is the hypotenuse. We'll deal with two right triangles: one for the top and one for the floor of the house.

Step 3: Analyzing the Triangle to the Top of the House

Given the angle of depression \( \alpha \) to the top of the house, the opposite side (vertical component from the tower to the top of the house) can be written using the tangent function: \( h = d \times \tan \alpha \), where \( h \) is the height from the tower's top to the house's top, and \( d \) is the distance along the ground from the tower to the house.

Step 4: Analyzing the Triangle to the Floor of the House

Similarly, the angle of depression \( \beta \) to the floor of the house leads to the equation \( 60 = d \times \tan \beta \), since the tower is 60 meters tall.

Step 5: Finding the Height of the House

Using the equations from Steps 3 and 4, the height of the house \( h_{house} = h_{floor} - h_{top} = d \times (\tan \beta - \tan \alpha ) \). We are given \( h_{house} = \frac{60 \sin (\beta - \alpha)}{x} \).

Step 6: Equating and Simplifying

Equate the expressions for \( h_{house} \): \( d \times (\tan \beta - \tan \alpha ) = \frac{60 \sin (\beta - \alpha)}{x} \). We know: \( d = \frac{60}{\tan \beta} \), substitute back to simplify: \( \frac{60}{\tan \beta} \times (\tan \beta - \tan \alpha) = \frac{60 \sin(\beta - \alpha)}{x} \).

Step 7: Identifying the Trigonometric Identity

Realize that \( \tan \beta - \tan \alpha = \frac{\sin(\beta - \alpha)}{\cos \alpha \cos \beta} \). Thus, \( x \) must simplify to \( \cos \alpha \cos \beta \).

Step 8: Conclusion

By the compatibility of the trigonometric simplifications, the correct answer is \( x = \cos \alpha \cos \beta \).

Key Concepts

Angles of Depression

When you're observing an object from above, the angle formed between the horizontal line of sight and the line to the object is called the angle of depression. In trigonometry, angles of depression are often used to solve real-world problems involving heights and distances.
For example, imagine you're standing on the top of a tower and looking down at a house. The angles of depression, denoted as \( \alpha \) and \( \beta \), respectively tell you the angle from the horizontal to the top and bottom of the house.
To visualize:

• Draw a horizontal line from your eyes directly outward from the tower.
• Draw another line downwards which creates an angle with this horizontal line.

These angles help in setting up the problem using trigonometric functions like tangent, which relates angles to the sides of right triangles.

Right Triangle Trigonometry

Right triangle trigonometry is the study of the relationships between the angles and sides of right triangles. In our exercise, two right triangles are pivotal: one involving the top of the house and another with its base floor.
Here are some important aspects:

• Right Triangle Definition: A triangle where one angle is exactly 90 degrees.
• Hypotenuse: The longest side opposite the right angle.
• Opposite & Adjacent: Relative to an angle, the opposite side is directly across, while the adjacent is beside it.

This is crucial for determining the vertical differences using angles of depression. For instance, the tangent function, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), allows for solving the problem by relating the height of the house to the ground distance using angles.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for every angle. In solving our exercise, recognizing and applying these identities is essential.
Consider the identity: \[\tan \beta - \tan \alpha = \frac{\sin(\beta - \alpha)}{\cos \alpha \cos \beta}\]This identity simplifies part of the problem to connect the angles of depression with the given height ratio expression.
By understanding such identities, you can transform complex expressions into simplified forms, leading to easier solutions.

• They help in converting between different trigonometric functions.
• They provide a way to compute unknown quantities when direct calculation isn't possible.

Hence, these identities not only support solving this specific problem but underpin many trigonometric applications in various disciplines.