Problem 3 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Three poles whose feet \(A, B, C\) lie on a circle subtend angles \(\alpha, \beta, \gamma\) and respectively, at the centre of the circle. If the height of the poles are in A.P, then \(\cot (\alpha), \cot (\beta), \cot (\gamma)\) are in (a) \(\mathrm{A} . \mathrm{P}\) (b) G.P (c) H.P (d) None

Step-by-Step Solution

Answer

The cotangents of the angles, that is \(\cot(\alpha)\), \(\cot(\beta)\), and \(\cot(\gamma)\), are in a Harmonic Progression (H.P). The correct option is (c) H.P.

Step 1: Visualize the scenario

Imagine you have a circle with a centre. From this centre, envision three lines extending to touch the circumference of the circle at points \(A\), \(B\), and \(C\). These lines represent the heights of the poles. Mark angles \(\alpha\), \(\beta\), and \(\gamma\) at the centre subtended by these lines.

Step 2: Apply trigonometric principles

We are given the heights are in an arithmetic progression, it means there are numbers \(h\), \(h+d\), \(h+2d\) respectively for \(A\), \(B\), \(C\). We know from trigonometry that \(\cot(\theta) = \frac{base}{perpendicular}\). Here the base is the radius of the circle and suppose it is \(r\). Therefore, the cotangents of the angles are \(\cot(\alpha) = \frac{r}{h}\), \(\cot(\beta) = \frac{r}{h+d}\), and \(\cot(\gamma) = \frac{r}{h+2d}\).

Step 3: Determine the sequence type

Now, in order to determine the type of sequence in which the cotangents of the angles are, we need to put the cotangents in terms of a common variable. By doing so, we get \(\cot(\alpha) = r* (1/h)\), \(\cot(\beta) = r*(1/(h+d))\), and \(\cot(\gamma) = r* (1/(h+2d))\). The reciprocals of the terms form an arithmetic progression, therefore, the terms form a harmonic progression (H.P).

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P) is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. For example, the sequence 2, 4, 6, 8, is an arithmetic progression with a common difference of 2. This means that the difference between any two successive terms is always the same.

In the problem provided, the heights of the poles are in an arithmetic progression:

• The first pole's height is denoted by a number, say \(h\).
• The second pole's height is \(h+d\), where \(d\) is the common difference.
• The third pole's height is \(h+2d\).

The arithmetic progression helps in understanding the linear increase in the heights of the poles, which is crucial for analyzing the sequence involved with their subtended angles.

Harmonic Progression

A harmonic progression (H.P) is a sequence of numbers derived from the reciprocals of an arithmetic progression. If the sequence \(a, a+d, a+2d, \ldots\) is an arithmetic progression, then its corresponding harmonic progression is \( \frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \ldots\).

In this exercise, the reciprocals of the heights of the poles that are in A.P lead us to form a harmonic progression:

• For height \(h\), the reciprocal is \(\frac{1}{h}\).
• For height \(h+d\), the reciprocal is \(\frac{1}{h+d}\).
• For height \(h+2d\), the reciprocal is \(\frac{1}{h+2d}\).

It directly implies that the cotangents \( \cot(\alpha), \cot(\beta), \cot(\gamma)\), by the problem's transformation, form an H.P. This is illustrated once we understand how these reciprocals can form such progression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. One key identity used in this exercise is for cotangent:\[\cot(\theta) = \frac{\text{adjacent side}}{\text{opposite side}} \]
This particular formula applies when circles are involved by using the circle's radius as the 'adjacent side'. In the problem, the radius \(r\) is used as the base when calculating for cotangents:

• \(\cot(\alpha) = \frac{r}{h}\)
• \(\cot(\beta) = \frac{r}{h+d}\)
• \(\cot(\gamma) = \frac{r}{h+2d}\)

Understanding this key identity enables students to grasp why the cotangents are being expressed in terms of the reciprocals of the heights, further leading to identifying their sequence nature.

Circle Geometry

Circle geometry is a branch of geometry that deals with the properties and measures related to circles. One fundamental aspect in this exercise is understanding how angles are subtended at the center of the circle by lines drawn to points on the circumference.

Imagine:

• There's a circle with its center, and from the center, lines are drawn to points \(A\), \(B\), and \(C\) on the circumference.
• These lines become the heights \(h, h+d, h+2d\), and they create angles \(\alpha, \beta, \gamma\) at the circle's center.

Circle geometry helps us visualize and comprehend the alignment and relation between the circle’s angles and the line lengths (here, heights of poles). This understanding is crucial when applying trigonometric identities and forming sequences from the given angle attributes.