Problem 9 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If in a \(\Delta A B C, \cos A+2 \cos B+\cos C=2\), then \(a, b\) \(c\) are in (a) A.P. (b) G.P. (c) H.P. (d) None

Step-by-Step Solution

Answer

The sides \(a\), \(b\) and \(c\) do not hold any relation. Thus, the correct answer is (d) None.

Step 1: Recall the Properties of Triangles and Cosine

The first thing to consider is that in any triangle \(\Delta ABC\), \(a\), \(b\) and \(c\) are sides opposite to angles \(A\), \(B\), and \(C\) respectively. Moreover, the sum of the angles in a triangle is always \(\pi\) radians, which translates to \(A + B + C = \pi\).

Step 2: Manipulate the Given Equation

We have the equation \(\cos A + 2\cos B + \cos C = 2\). Using the cosine rule and replacing \(B\) and \(C\) in terms of \(A\), we get \(\cos A + 2\cos(\pi - C - A) + \cos(\pi - A - B) = 2\). The cosine rule states that \(\cos(x - y) = \cos x \cos y + \sin x \sin y\), and \(\cos(\pi - x) = -\cos x\). So the equation becomes \(\cos A - 2(\cos A \cos C + \sin A \sin C) - (\cos A \cos B + \sin A \sin B) = 2\).

Step 3: Simplify the Equation

Simplifying the equation further and assuming that all terms involving \(A\), \(B\), and \(C\) are zero since they are constants, the left hand side then simplifies to -2\(\cos A (\cos B + \cos C) + 2\sin A (\sin B + \sin C) = -2\cos A\). Equating it to 2 (the right-hand side), we get \(-2\cos A = 2\), consequently, \(\cos A = -1\).

Step 4: Find the Relationship Between the Sides

Given \(\cos A = -1\), it means angle \(A = \pi\) radians or \(180^\circ\). In a triangle, if one angle is \(180^\circ\), the other two angles must be \(0^\circ\), hence \(B = C = 0^\circ\). Now, we are required to find the relationship between the sides, which can be done using the sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Since \(\sin 0 = 0\) and \(\sin 180 = 0\), substitute \(A = 180^\circ\), \(B = C = 0^\circ\) into this equation. We get \(\frac{a}{0} = \frac{b}{0} =\frac{c}{0}\), which is undefined, meaning the three sides do not hold any common ratio or common difference. Hence, \(a\), \(b\) and \(c\) are not in AP, GP or HP.

Key Concepts

Cosine Rule

The Cosine Rule is an essential trigonometric tool for solving triangles, especially when dealing with non-right triangles. It connects the lengths of a triangle's sides with the cosine of one of its angles. The formula is given by \(c^2 = a^2 + b^2 - 2ab \cdot \cos C\), where:

• \(a, b, \) and \(c\) are the lengths of the sides of the triangle.
• \(C\) is the angle opposite to side \(c\).

This rule is particularly useful when you know two sides and the included angle, or all three sides of a triangle. It helps determine any unknown angles or side lengths. In problems like the one given, the cosine rule might be manipulated to find relationships among angles and sides by substituting known values and solving the resulting equations.

Triangle Angle Sum

In any triangle, the sum of all interior angles is always \(180^\circ\) or \(\pi\) radians. This property is crucial in solving many geometric problems. Let's break it down:

• Angle \(A + B + C = 180^\circ\).
• Knowing two angles allows you to find the third.
• This rule applies uniformly to all types of triangles – whether scalene, isosceles, or equilateral.

Used alongside trigonometric identities and rules, the angle sum property serves as a foundational element in proving other aspects of triangle geometry, like determining if sides are in a specific progression, as attempted in the solution above.

Arithmetic Progression

An Arithmetic Progression (AP) is a sequence where each term is obtained by adding a constant difference to its preceding term. If the sides of a triangle were to be in AP, this would mean:

• \(b - a = c - b\)
• The common difference is equal for all pairs of consecutive terms.

For triangle sides to form an AP, the relationship derived from angle properties and lengths would consistently reflect this same difference. As shown in the original exercise solution, certain mathematical configurations of angles or sides might not lead to an arithmetic relationship, hence not forming an AP.

Geometric Progression

A Geometric Progression (GP) involves a sequence where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. If triangle sides were considered to form a GP:

• \(\frac{b}{a} = \frac{c}{b}\)
• Every subsequent side length is a product of the previous side and the common ratio.

Exploring whether sides of a triangle can be in GP involves assessing angle properties and applying trigonometric rules. The solution pointed out undefined results which indicate the sides of the triangle do not form such a progression, which frequently happens when angle combinations lead to invalid or illogical scenarios.

Harmonic Progression

In a Harmonic Progression (HP), the reciprocals of the terms form an arithmetic progression. This means that if sides \(a, b,\) and \(c\) were in HP:

• The reciprocal sequence \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) forms an AP.
• There’s a common difference between the reciprocals.

Determining an HP relationship in triangle sides encompasses initially transforming sides into their reciprocals and expecting a steady increment akin to an AP. As observed in the exercise solution, undefined behavior from the mathematical approach leads to the conclusion that the sides do not constitute a harmonic series. Using trigonometric rules effectively illustrates why a harmonic relation is not feasible in the given problem.