Problem 51 - A Complete Resource Book in Mathematics for JEE Main 2017

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 answer is (C): \(\tan^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)}\right)\).

Step 1: Sketch the Problem

First, let's visualize the problem. Consider a right-angled triangle formed by a point on the hill and the top of the flagstaff. The angle of elevation is \(\alpha\) when observed from one point, and \(\beta\) when observed from a point one meter higher, closer to the hilltop. The height of the flagstaff is \(h\).

Step 2: Use Trigonometric Relationships

The tangent of an angle of elevation in a right triangle is the ratio of the opposite side to the adjacent side. From the point where the angle of elevation is \(\alpha\), the relation is: \(\tan \alpha = \frac{h}{d}\), where \(d\) is the horizontal distance from the point to directly under the flagstaff.

Step 3: Set Up Second Equation

From a point one meter closer, the angle of elevation is \(\beta\). The relationship becomes \(\tan \beta = \frac{h}{d-a}\), where \(d-a\) is now the horizontal distance to the new point. \(a\) is the horizontal distance moved.

Step 4: Calculate Difference

To find the difference between the two angles' equations: \(\tan \beta - \tan \alpha = \frac{h}{d-a} - \frac{h}{d}\). Simplify using the tangent subtraction formula, \(\tan(\beta-\alpha) = \frac{a}{a'}\), where \(a'\) is the horizontal change needed based on hill inclination and \(a\).

Step 5: Express Hill Inclination

The tangent of the hill's inclination, \( \theta\), is \( \tan \theta = \frac{a \sin \alpha \sin \beta}{h \sin(\beta - \alpha)} \). This expression comes from expressing the relations above, treating \(a\) over \(h\) as the same measure over the hypotenuse.

Step 6: Final Answer for Inclination

Now, applying \( \theta = \tan^{-1}(\tan \theta) \), select the answer choice that matches the derived expression. This corresponds to option (C): \( \tan ^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin (\beta-\alpha)}\right) \).

Key Concepts

Angle of Elevation

The angle of elevation is a key concept in trigonometry and is crucial in understanding certain real-world problems, like the one involving the flagstaff on a hill. Imagine standing on the ground and looking up at a tall object ahead of you, such as a flagpole or a mountain. The angle your line of sight makes above the horizontal line at your eye level is called the angle of elevation.
To visualize this, think of your eyes tracing an imaginary line upwards towards the top of the object. This upward-looking angle, formed above the horizontal, is what we term the "angle of elevation." Remember: as you move closer to the object, the angle of elevation typically increases, while moving further away results in a smaller angle.
Here are some ways the angle of elevation is used:

• Measuring the height of inaccessible objects, like mountains or buildings.
• Calculating the distance to objects when their height is known.
• Used in various engineering and aviation tasks to ensure accuracy in construction and navigation.

Understanding the angle of elevation sets the stage for applying trigonometric functions and solving elevation and distance-related problems.

Right-Angled Triangle

In trigonometry, the right-angled triangle is the foundational shape that helps us understand and apply various trigonometric functions. The problem of calculating the hill's inclination revolves around understanding the properties of such a triangle. A right-angled triangle has one 90-degree angle. The two sides forming this right angle are known as the base and the height. The longest side, opposite the right angle, is the hypotenuse.
In real-world contexts, like the flagstaff problem, this triangle helps simplify complex situations. Here are some important aspects:

• The side opposite the angle of elevation is typically the height of the object being measured.
• The base is the portion of the landscape or distance on the ground.
• The hypotenuse runs from the observer's eye to the top of the object being observed.

Remember: applying the principles of right-angled triangles allows you to use trigonometric ratios, like sine, cosine, and tangent, to solve for unknown variables, such as distances or heights. This triangle's properties are elegantly intertwined with the angle of elevation, making them indispensable for such calculations.

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. In the context of the angle of elevation, it becomes particularly useful when solving for distances and heights in a right-angled triangle. The tangent of an angle in such a triangle is defined by the ratio of the opposite side to the adjacent side.
Mathematically, if you have an angle \( \theta \), its tangent is expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]In our flagstaff example, this means:

• If you know the height of the flagstaff and the angle of elevation, you can determine the ground distance from the viewing point to the base of the flagstaff.
• This function is especially significant when you have measurements like these, and it directly supports solving real-world problems by linking angles and lengths.

The power of the tangent function lies in its simplicity and practicality, turning angles into measurable insights about distances, critical for construction, navigation, and various scientific applications.