Problem 10 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Let \(a, b\) and \(c\) be the sides of a \(\Delta A B C\). If \(a^{2}, b^{2} \& c^{2}\) are the roots of the equation \(x^{3}-P x^{2}+Q x-R=0\), where \(p, q\) and \(R\) are constants, then find the value of \(\frac{\cos A}{a}+\frac{\cos B}{h}+\frac{\cos C}{c}\), in terms of \(P, Q\) and \(R\).

Step-by-Step Solution

Answer

The final result is \(-\frac{P}{R}\)

Step 1: Cosine rule for angle A

The Cosine Rule for angle A is given by \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)

Step 2: Cosine rule for angle B

Apply the same rule for angle B, we get \(\cos B = \frac{a^2 + c^2 - b^2}{2ac}\)

Step 3: Cosine rule for angle C

Lastly, for angle C, the Cosine Rule gives us \(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\)

Step 4: Adding up the cosines

Add up \(\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c}\), by putting in the results from previous steps, we get \(\frac{(b^2 + c^2 - a^2)}{2abc} + \frac{(a^2 + c^2 - b^2)}{2abc} + \frac{(a^2 + b^2 - c^2)}{2abc}\)

Step 5: Resolving addition

When simplify, it is noticed that a, b, and c get cancelled out while squared terms add up: \( \frac{c^2 + b^2 + a^2}{2abc} = \frac{2P}{2abc}\)

Step 6: Final result

It simplifies to \(\frac{P}{abc}\). The product of roots of a cubic equation taken three at a time is equal to \(-R\) (because x³ has negative coefficient). Thus, \(abc = - R\) and our final answer is \(-\frac{P}{R}\)

Key Concepts

Cosine Rule

The Cosine Rule is an essential tool in trigonometry, especially when dealing with triangles that are not right-angled. It helps to find a side of a triangle when two sides and the enclosed angle are known or to find an angle when all sides of a triangle are known. The formula is given by:
\[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]
This version of the rule is used to find the cosine of angle A in triangle \( \Delta ABC \), where \( a, b, \) and \( c \) are the sides opposite to the angles A, B, and C, respectively.

• To determine \( \cos A \), the length of side \( a \) is subtracted from the sum of squares of the other two sides \( b \) and \( c \), then divided by twice the product of those two sides.
• Similarly, \( \cos B \) is found using \( \frac{a^2 + c^2 - b^2}{2ac} \), and \( \cos C \) using \( \frac{a^2 + b^2 - c^2}{2ab} \).

By reinforcing the relationship between the triangle’s sides and angles, the Cosine Rule forms a bridge to solving many trigonometric problems.

Cubic Equation Roots

Cubic equations are polynomial equations of degree three, commonly structured as \( x^3 - Px^2 + Qx - R = 0 \). Understanding how to extract the roots of a cubic equation is crucial, especially when they relate to geometric problems like triangles.
The given problem states that the roots of the cubic equation are \( a^2, b^2, \) and \( c^2 \) for the triangle's sides \( a, b, \) and \( c \).

• The sum of roots, based on the equation, is \( P = a^2 + b^2 + c^2 \), which is a reflection of the coefficients of the equation.
• The product of roots, derived from cubic properties, is \( abc = -R \) due to the equation's structure.

Understanding these relationships allows for deeper insights into both algebraic manipulations and geometric interpretations. The ability to transpose between these forms is a powerful tool for solving complex problems.

Triangle Sides

In trigonometry, the sides of a triangle are not just measurements; they are components that relate to various algebraic and trigonometric functions. In the context of our problem, each side of the triangle is linked intricately with a root of a cubic equation.
When dealing with triangle sides in a trigonometric and algebraic context:

• The sides \( a, b, \) and \( c \) are related to their squared values in a cubic equation, as squares form the roots of the equation.
• These sides also work as denominators when using the Cosine Rule, which signifies the interplay between the side's lengths and the angles opposite them.

By solving an expression like \( \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} \), students can see how algebraic identities and triangles' geometric properties overlap. Understanding the relationship between the angles and the triangle sides is critical for navigating both standalone trigonometric problems and their integration into algebraic systems.