Problem 51 - 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 \(\mathrm{h}\) 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 inclination of the hill is (A) \( \sin^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin(\beta - \alpha)}\right) \).

Step 1: Set Up the Problem with Right Triangles

Consider two right triangles representing the scenario. Let the point on the hill from where the first observation is made be A, and the top of the flagstaff be B. Let C be the point vertically below B at the base of the flagstaff on the hill. D is a point 1 meter closer to the flagstaff at the same incline of the hill. We need to find the relationship between these triangles using trigonometric identities.

Step 2: Establish Initial Elevation Equation

From point A, the angle of elevation to the top of the flagstaff is \( \alpha \). The triangle ABC gives us: \( \tan \alpha = \frac{h}{x} \), where \( x \) is the horizontal distance from point A to C down the incline of the hill.

Step 3: Establish Second Elevation Equation

From point D, 1 meter closer to the top than A, the angle of elevation is \( \beta \). The new triangle DBC gives us: \( \tan \beta = \frac{h}{x-a} \), where \( x-a \) is the horizontal distance from point D to C. Here, \( a = 1 \) meter.

Step 4: Use Trigonometric Identity to Relate Angles

We have two equations from right triangles: \( \tan \alpha = \frac{h}{x} \) and \( \tan \beta = \frac{h}{x-a} \). Solving these, we get two relationships. Subtract them to find \( x \) in terms of \( a \), \( h \), and angles \( \alpha \), \( \beta \).

Step 5: Derive the Hill's Inclination

The difference in tangents: \( \tan \beta - \tan \alpha = \frac{h}{x-a} - \frac{h}{x} = \frac{ha}{x(x-a)} \). This simplifies to the form: \( \frac{a \sin \alpha \sin \beta}{h \sin(\beta - \alpha)} \), where \( a = 1 \) meter, derived from trigonometric identities. This expression gives the sine of the angle of inclination of the hill.

Step 6: Conclusion with the Inclination Formula

The inclination of the hill \( \phi \) to the horizontal, which corresponds to the derived expression \( \sin \phi = \frac{a \sin \alpha \sin \beta}{h \sin(\beta - \alpha)} \), can be found using the inverse sine function: \( \phi = \sin^{-1}\left(\frac{a \sin \alpha \sin \beta}{h \sin(\beta - \alpha)}\right) \).

Step 7: Verify Answer with Given Options

Compare the derived formula with the given options. Option (A) matches exactly, confirming that it is the correct answer.

Key Concepts

Angle of Elevation

When you're standing at the base of a hill and looking up at something tall, like a flagpole, you're observing an angle of elevation. This angle is the angle between your line of sight and the horizontal line from where you stand. It tells you how much you need to look upwards to see the top of the object.

In the exercise, this angle of elevation is represented by two different values because one is measured from point A and the other is from point D, a meter closer to the flagstaff. When standing at point A, the angle of elevation is observed as \( \alpha \), and from point D, it is \( \beta \).

• The angle of elevation helps in setting up right triangles, as it is crucial for applying trigonometric identities.
• Understanding this concept is key for solving problems involving height and distances seen from an angle.

Right Triangles

Right triangles are special triangles that have one angle exactly 90°. They are fundamental in trigonometry because their properties and relationships are predictable and easy to work with using trigonometric ratios like sine, cosine, and tangent.

In this problem, two right triangles are involved: \( \triangle ABC \) and \( \triangle DBC \). Point B is at the top of the flagstaff, Point C is at the base where the flagstaff meets the ground, and Points A and D are on the hillside.

• These triangles help visualize and form equations needed to find unknown measurements.
• The side opposite the angle of elevation in these triangles is the height of the flagstaff \( h \), and the adjacent side corresponds to the horizontal distances \( x \) or \( x-a \).

By understanding the structure of these triangles, you can apply the tangent function, which is suitable here due to the involvement of opposite and adjacent sides.

Trigonometric Identities

Trigonometric identities are formulas that relate the angles and sides of triangles. In this particular problem, we use the identities that involve tangent to relate the angles of elevation to the sides of the right triangles.

Key concepts of trigonometric identities used:

• The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
• For angles \( \alpha \) and \( \beta \) from Point A and D, you use the tangent identity: \( \tan \alpha = \frac{h}{x} \) and \( \tan \beta = \frac{h}{x-a} \).
• The difference identity for tangents is utilized here: \( \tan \beta - \tan \alpha = \frac{ha}{x(x-a)} \).

These identities enable the simplification of the trigonometric equations to find the inclination \( \phi \) of the hill. Knowing the angle of inclination provides valuable information about the slope, which is essential in many fields such as construction, navigation, and even sports like skiing and hiking.