Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A tower \(\mathrm{AB}\) leans towards west making an angle \(\alpha\) with the vertical. The angular elevation of \(\mathrm{B}\), the top most point of the tower is \(\beta\), as observed from a point \(\mathrm{C}\) due east of \(\mathrm{A}\) at a distnace \(\mathrm{d}\) from \(\mathrm{A}\). If the angular elevation of \(\mathrm{B}\) from a point due east of \(\mathrm{C}\) at a distance \(2 \mathrm{~d}\) from \(\mathrm{C}\) is \(\gamma\), then

Step-by-Step Solution

Answer

\(\tan\gamma = \frac{1}{3}\tan\beta\) relates \(\gamma\) and \(\beta\).

Step 1: Understand the Geometry of the Situation

Visualize the scenario described in the problem. Imagine the leaning tower AB with the point B making an angle \(\alpha\) with the vertical. Point C is due east of point A, at a distance \(d\). From this point C, the top of the tower B forms an angular elevation \(\beta\). We need to find the elevation angle, \(\gamma\), when moved further to a point 2d from C, eastwards.

Step 2: Relate the Angles with the Triangle

The angles \(\alpha\) and \(\beta\) are connected through right triangle trigonometry involving the horizontal and vertical components of the tower. The height of the tower can be expressed as \(h = AB\sin\alpha\). The distance CA is \(d\).

Step 3: Use Trigonometry to Find Relations

From the point C, the tangent of the angle \(\beta\) can be expressed as \(\tan\beta = \frac{h}{d}\). Also, for the new point at distance \(3d\) (due east of C by additional distance \(2d\)), the tangent of the angle \(\gamma\) is \(\tan\gamma = \frac{h}{3d}\).

Step 4: Solve for \(\gamma\) Using Trigonometric Relationships

Since \(\tan\beta = \frac{h}{d}\) and \(\tan\gamma = \frac{h}{3d}\), we can use the fact that \(\tan\gamma = \frac{1}{3}\tan\beta\) to find \(\gamma\) in terms of \(\beta\), which does not directly give the angle \(\gamma\) but shows its relation with known quantities.

Key Concepts

Angular Elevation

Angular elevation refers to the angle between the horizontal line from the observer's eye to a point above the observer's sight. It is commonly used in trigonometry to determine heights and distances, especially when the object being observed is on the ground, like a tower.
Angular elevation can be visualized as the angle formed when you tilt your head upward from the horizon to see the top of a building or a mountain. In the given exercise, the angular elevation \(\beta\) is the angle observed when looking at the topmost point B of the leaning tower from a point at the base.Key points to remember about angular elevation:

• It is always measured from a horizontal line.
• It helps to determine the height of an object through trigonometric relationships.
• It is a crucial part of trigonometric problems involving heights and distances.

Understanding angular elevation helps in connecting the concept of looking upward at an angle to calculate other dimensions involving the structure in question.

Trigonometric Relationships

Trigonometric relationships play a crucial role in solving problems with angles and distances. These relationships involve functions like sine, cosine, and tangent, which help describe various properties of triangles, especially right triangles. They are fundamental in determining length, height, and angle measurements using given values.
For instance, given the formula \(\tan\beta = \frac{h}{d}\), it reflects the relationship between the angle \(\beta\), the height \(h\) of the tower, and the distance \(d\) from the observer to the base of the tower. This basic trigonometric setup uses tangent, a function that relates the opposite side to the adjacent side in right triangles.Important trigonometric functions include:

• Sine (sin): Ratio of the opposite side to the hypotenuse.
• Cosine (cos): Ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Ratio of the opposite side to the adjacent side.

In the context of the original exercise, using trigonometric relationships can help us understand how the angle \(\gamma\) relates to the angle \(\beta\) when moving further from the initial observation point.

Right Triangle Trigonometry

Right triangle trigonometry is a special branch of trigonometry focusing on triangles where one angle is exactly 90 degrees. This field of study is fundamental because it allows us to solve many geometric problems involving distance and angles.
In the scenario described, the leaning tower forms various right triangles with the horizontal plane and the line of sight to the observers. These triangles are crucial in applying trigonometric ratios. For instance, knowing how to compute the tangent of angles involving height and several distances simplifies finding unknown measures, like additional angles or lengths, such as \(\gamma\).
Key applications of right triangle trigonometry include:

• Using the Pythagorean theorem to solve for sides.
• Applying trigonometric ratios to find unknown angles or lengths.
• Understanding the relationship between angles and side lengths in optimization problems.

Understanding right triangle trigonometry is indispensable for tackling exercises like these where you have to visualize the scenario and apply trigonometric principles accurately to find solution.