Problem 42 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a triangle \(A B C\), if the sides \(a, b, c\) be in A.P., prove that \(\cos A \cdot \cot \left(\frac{A}{2}\right), \cos B \cdot \cot \left(\frac{B}{2}\right), \cos C \cdot \cot \left(\frac{C}{2}\right)\) are also in A.P.

Step-by-Step Solution

Answer

We applied trigonometric identities, Sine and Cosine Rule to simplify the given sequence \(\cos A \cdot \cot \left(\frac{A}{2}\right), \cos B \cdot \cot \left(\frac{B}{2}\right), \cos C \cdot \cot \left(\frac{C}{2}\right)\), to prove that it follows the progressional pattern of an Arithmetic Progression.

Step 1: Understanding the Concept of Arithmetic Progression (AP)

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant. This means if \(a, b, c\) are in AP, then \( b-a = c-b \). Using this information, one can rewrite \(b\) as \((a+b)/2\) and \(c\) as \(2b - a\)

Step 2: Use Cosine and Cotangent Trigonometric Identities

Using the identity \(\cos {2B} = 2(\cos^2 {B}) - 1\) and \(\cot {(\frac{B}{2})} = \frac{1 + \cos {B}}{\sin {B}}\), we can substitute them on the sequence. This will yield: \(\cos {A} \cdot \cot \left(\frac{A}{2}\right) = a \cdot \left( \frac{1 + \cos {A}}{\sin {A}} \right)\), \(\cos {B} \cdot \cot \left(\frac{B}{2}\right) = b\cdot \left( \frac{1 + \cos {B}}{\sin {B}} \right)\), and \(\cos {C} \cdot \cot \left(\frac{C}{2}\right) = c \cdot \left( \frac{1 + \cos {C}}{\sin {C}} \right)\)

Step 3: Use the Sine Rule and the Cosine Rule

We don't have the values for angles \(A,B,C\). Hence, we will use the sine rule and cosine rule. According to the sine rule, \(\sin B = \frac{b \sin A}{a}\). The cosine rule states \(\cos B = \frac{a^2 + b^2 - 2ab \cos A}{2ab} \). Using these rules, the given sequence shall be simplified and expressions shall be rewritten as \({a^2}\), \({b^2}\) and \({c^2}\)

Step 4: Prove that these are in Arithmetic Progression

Now we can prove that \({a^2}\), \({b^2}\) and \({c^2}\) are in AP using the definition from Step 1. This simplifies to \({b^2} - {a^2} = {c^2} - {b^2}\), which therefore fulfills the criteria of arithmetic progression.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, and tangent. They are used to simplify complex trigonometric expressions and solve equations involving angles. Key identities include:

• Pythagorean identities: such as \( \sin^2\theta + \cos^2\theta = 1 \)
• Double angle identities: such as \( \cos(2\theta) = 2\cos^2\theta - 1 \)
• Half angle identities: such as \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1+\cos\theta}{2}} \)

In solving triangle problems, trigonometric identities help relate the sides and angles of the triangle by converting them into known patterns. This simplification is crucial in the step-by-step solution where cotangent and cosine identities are used to express the given conditions in a manageable format for further simplification.

Sine Rule

The sine rule, also known as the law of sines, is a useful relationship in any triangle that relates the ratios of the sides to the sines of their opposite angles. The rule is expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This means if you know some of the angles and sides in a triangle, you can use the sine rule to find unknown lengths or angles. In the given problem, although the angles \(A, B, C\) are initially unknown, the sine rule allows us to express these angles in terms of the given sides, thereby revealing relations between the sides and angles. This step is essential in transforming the expressions to check for arithmetic progression.

Cosine Rule

The cosine rule is another fundamental tool in trigonometry, especially useful when dealing with non-right angled triangles. It generalizes the Pythagorean theorem and holds for any triangle, helping to find unknown sides or angles. The cosine rule is often stated as:\[ c^2 = a^2 + b^2 - 2ab\cos C \]This equation helps express the third side of a triangle if we know the length of two sides and the measure of the included angle. Similarly, it can be rearranged to solve for an angle if all side lengths are known:\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]In the problem at hand, the cosine rule is used to understand the relation between angle \(B\) and the side lengths \(a, b, c\). This aids in rewriting the expressions in terms of squares of side lengths, an intermediate step crucial for demonstrating the arithmetic progression of the derived expressions.