Problem 2 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Sides of a triangle \(A B C\) are in A.P. If \(a<\min \\{b, c\\}\), then \(\cos A\) may be equal to (a) \(\frac{3 c-4 b}{2 a}\) (b) \(\frac{3 c-4 b}{2 c}\) (c) \(\frac{4 c-3 b}{2 b}\) (d) \(\frac{4 c-3 b}{2 c}\)

Step-by-Step Solution

Answer

(a) \( \frac{3 c-4 b}{2 a} \)

Step 1: Define the sides of the triangle

Since the sides of the triangle are in arithmetic progression and 'a' is smaller than 'b' and 'c', it's logical to assume that the sides of the triangle are 'a', 'a+d', and 'a+2d' where 'd' is the common difference. Here, 'b = a + d' and 'c = a + 2d'.

Step 2: Use the cosine rule

Next, apply the cosine rule which states that \( cosA = \frac{b^2 + c^2 - a^2}{2bc} \). Under the defined conditions, replace 'b' with '(a + d)' and 'c' with '(a + 2d)'.

Step 3: Simplify the equation

After applying the cosine rule and performing simplification, the equation becomes \( cosA = \frac{3a^2 + 4ad}{2a^2 + 4ad}. \) Divide numerator and denominator by \(4ad\) for further simplification.

Step 4: Compare with the provided options

The simplified equation for cos A is \( cos A = \frac{3c - 4b}{2a} \). This matches with option (a). Therefore, when sides of a triangle \(A B C\) are in A.P. and \(a<\min \{b, c\}\), \( \cos A = \frac{3 c-4 b}{2 a} \).

Key Concepts

Arithmetic Progression in Triangle Sides

Understanding the concept of Arithmetic Progression (AP) in the context of triangle sides can be crucial for solving a variety of geometric problems. An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant difference, known as the common difference, to the previous term. When we talk about the sides of a triangle being in AP, we imply that the lengths of the sides form such a sequence.

For instance, if we are given that the sides of triangle ABC are in AP, and we know that side a is the smallest, we can denote the sides as a, a+d, and a+2d - where d is the common difference. This notation is incredibly helpful for applying theorems and rules that intersect with the properties of AP to derive relationships between the sides of the triangle and its angles.

Cosine Rule Application

The Cosine Rule, also known as the Law of Cosines, is an essential tool in triangle geometry, especially when dealing with non-right-angled triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is given by \( c^2 = a^2 + b^2 - 2ab\cos(C) \) where a, b, and c are the sides of the triangle, and C is the angle opposite side c.

If we rearrange this formula to solve for \( \cos(C) \) we get \( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \). The application of the Cosine Rule becomes particularly interesting when we have sides in arithmetic progression, as this imposes an additional structure on the side lengths, enabling us to further transform the formula into a more specific form that can exhibit patterns or lead to simplifications when analyzing the triangle's properties.

Trigonometric Ratios of Triangles

Triangles, the simplest polygons, are not only fundamental shapes in geometry but also in trigonometry. The trigonometric ratios—such as sine, cosine, and tangent—are functions that relate the angles of a triangle to the ratios of its sides. Applying these ratios to triangles enables us to solve problems involving angles and distances without the need to directly measure them.

In the context of non-right triangles, the cosine of an angle is of particular interest as it can be used to find unknown side lengths or angles when certain other measurements are known. It is also part of the Cosine Rule, which is a form of the trigonometric ratios extended to apply to any triangle, not just right-angled ones. Hence, when sides of a triangle are in AP and we aim to find the cosine of one of the angles, we deploy both our understanding of how the sides relate in AP and how trigonometric ratios like the cosine can be expressed in terms of side lengths.