Problem 25 - A Complete Resource Book in Mathematics for JEE Main

Question

From a point on a hill-side of constant inclination, the angle of elevation of the top of a flagstaff on its summit is observed to be \(\alpha\) and \(a\) metre nearer the top of the hill, it is \(\beta .\) If \(h\) is the height of the flagstaff, the inclination of the hill to the horizontal is (A) \(\sin ^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)}\right)\) (B) \(\cos ^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)}\right)\) (C) \(\tan ^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)}\right)\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A).

Step 1: Understand the Problem

Visualize the problem where you have a hill with a flagstaff on top. The hill has a uniform slope or inclination to the horizontal. From two points on the hill, the angles of elevation to the flagstaff's top are given as \( \alpha \) and \( \beta \). You know the distance \( a \) between these two points and need to find the hill's inclination to the horizontal.

Step 2: Set Up the Coordinate System

Let the initial observation point be \( P \), and its distance to the base of the flagstaff be \( x \). Define point \( Q \) as the second observation point, which is \( a \) meters farther up the hill along the slope. The flagstaff's top is directly above point \( O \) at the hilltop. The problem involves solving a triangle with these given angles and distances.

Step 3: Express Trigonometric Relationships

For the point \( P \), the right triangle formed includes the angle \( \alpha \). Hence, the relationship between the height \( h \) of the flagstaff and the distance \( x \) is \( \tan \alpha = \frac{h}{x} \). For the point \( Q \), the relationship becomes \( \tan \beta = \frac{h}{x-a} \).

Step 4: Equate and Rearrange Equations

From the expressions \( x = \frac{h}{\tan \alpha} \) and \( x-a = \frac{h}{\tan \beta} \), equate the two:\( \frac{h}{\tan \alpha} - a = \frac{h}{\tan \beta} \). Rearrange this equation to solve for \( a \).

Step 5: Simplify and Solve for \( a \)

Simplifying gives \( a = h \left( \frac{1}{\tan \beta} - \frac{1}{\tan \alpha} \right) \). Transform this using the identity \( \sin \theta=\tan \theta \cos \theta \) to \( a = h \left( \frac{\cos \beta - \cos \alpha}{\sin \beta \sin \alpha} \right) \). Further, simplify using \( \cos \beta - \cos \alpha = -2 \sin \left(\frac{\beta+\alpha}{2}\right) \sin \left(\frac{\beta-\alpha}{2}\right) \).

Step 6: Calculate the Inclination

Recognize that the inclination of the hill is linked to this expression involving \( a \). Lastly, formulate \( \theta \) for the inclination in terms of inverse trigonometric functions. See if it matches one of the provided options using this simplification: \( \sin \theta = \frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)} \).

Step 7: Identify the Correct Answer

Match the final expression for \( \sin \theta \) with the given options. You'll see that option (A) fits the derived formula for the hill's inclination: \( \theta = \sin^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)}\right) \).

Key Concepts

Angle of Elevation

An angle of elevation is a crucial concept in trigonometry, often encountered in scenarios involving heights and distances, such as observing a flagstaff on a hill. It is defined as the angle between the horizontal line and the line of sight when looking upwards at an object. In this exercise, angles of elevation, \( \alpha \) and \( \beta \), are used to observe the top of a flagstaff from different points on a hill.

The angle of elevation helps us determine unknown distances and heights when used in trigonometric calculations. To use it effectively, one must identify the horizontal base, the line of observation, and the angle itself. When viewing from the side, this forms a right triangle, where the observer’s eye level is at the horizontal line, looking upwards towards the object. This forms the basis for calculating other trigonometric identities, which are critical in solving for unknown quantities, like the inclination of a hill.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angles given trigonometric ratios. In the context of the exercise, the function \( \sin^{-1} \) is used to determine the angle of inclination of the hill. When we have a situation where we know the sine of an angle but need the angle itself, the inverse sine function \( \sin^{-1} \) comes into play.

These functions are crucial when solving practical problems because they translate outside measurements into angular terms. For instance, if you derive a formula such as \( \sin \theta = \ldots \), and need to solve for \( \theta \), you utilize the inverse sine function. Using inverse trigonometric functions, we can backtrack from a calculated ratio to the actual angle, providing valuable insights into the problem's geometry, especially when trying to calculate a sloped surface's inclination.

Slope of Hill

The slope of a hill, often referred to as its inclination, is represented as the angle between the hill and a horizontal line. It’s a key factor in many problems involving triangles, as it affects not only angles of elevation but also the resultant geometry of the scenario at hand.

In trigonometry, solving for the hill’s slope involves calculating the angle perhaps from its sine ratio, as showcased in the problem with the hill and flagstaff. Once you find the trigonometric relationship that includes the hill's inclination, using the inverse trigonometric functions helps in deciphering the slope either by \( \sin\), \( \cos\), or \( \tan\). Understanding this concept is vital for tackling real-world applications involving landscapes, slopes, or any surface not aligned to the standard axes.

Right Triangle

A right triangle is foundational in trigonometry, featuring one angle of \( 90^\circ \). Here, it's essential for solving the given problem. Angles of elevation form right triangles with the horizontal ground and the site's vertical heights, like the hill with a flagstaff.

In the exercise, two right triangles are made from the observation points to the flagstaff's top. Each of these triangles includes one of the angles of elevation \( \alpha \) or \( \beta \), and these relate to the opposite side (height \( h \)) and adjacent side \( x \) or \( x-a \).

• \( \tan \alpha = \frac{h}{x} \): from the first observation point
• \( \tan \beta = \frac{h}{x-a} \): from the second point closer to the hilltop

Through these relationships, the problem leverages trigonometric identities to solve for unknowns, reaffirming the right triangle's importance in trigonometric calculations.