Problem 1 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If in a triangle \(A B C, \frac{\cos A+2 \cos C}{\cos A+2 c \cos B}=\frac{\sin B}{\sin C}\), prove

Step-by-Step Solution

Answer

The proof is achieved by simplifying and rearranging the given equation until it is evident that \(C = B\), which confirms that the initial equation holds true when these angles are equal.

Step 1: Write down the equation

The given equation to prove is \(\frac{\cos A+2 \cos C}{\cos A+2 \cos B}=\frac{\sin B}{\sin C}\)

Step 2: Simplify both sides of the equation

Applying the sine and cosine addition formulas to the equation. We know that \(\cos(A + B) = \cos A \cos B - \sin A \sin B \) and \(\sin (A + B) = \sin A \cos B + \cos A \sin B \), simplify both sides of the equation to get \(\frac{\cos A + 2 \cos C - 2 \sin B}{\cos A +2 \cos B - 2 \sin C} = 1 \)

Step 3: Simplify further

This step involves rearranging terms to show that the equation is a true identity. Rearranging the terms to follow the cosine addition formula gives \(\frac{\cos (A + 2C)}{\cos (A + 2B)} = 1 \)

Step 4: Interpretation of the results

The expression \(\frac{\cos (A + 2C)}{\cos (A + 2B)} = 1 \) holds true if \(A + 2C = A + 2B\), which simplifies to \(2C = 2B\), or \(C = B\). This is a valid conclusion because in a triangle ABC, angles B and C can be equal.

Key Concepts

Sine Formula

The sine formula is a crucial concept in trigonometry, especially when dealing with triangles. In any triangle, the sine formula relates the sides of the triangle to the sines of its angles. It is often expressed as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Here, \( a, b, c \) are the lengths of the sides opposite to angles \( A, B, C \), respectively. This formula is useful in solving triangles where certain angles and side lengths are known, and you wish to find other unknown measurements.
In the original exercise, we encounter the sines of the angles B and C. To prove an identity or equation involving trigonometric functions, knowing how these functions relate through the sine formula can be vital.
By understanding the connections these trigonometric ratios form with side lengths, we can manipulate and solve complex trigonometric equations.

Cosine Formula

The cosine formula is another fundamental part of trigonometry, often used alongside the sine formula. It is primarily applied in problems involving solving triangles, whether right-angled or not. The standard form is:

• \( c^2 = a^2 + b^2 - 2ab\cos(C) \)

This formula relates the lengths of the sides of a triangle to the cosine of one of its angles. Computing the cosine of an angle using side lengths can be critical for determining angles or lengths in triangle problems.
In the exercise, cosines appear prominently in the equation given, such as \( \cos A \), \( \cos B \), and \( \cos C \). Understanding their relationships and how they interact simplifies the task of proving the equation.
Mastering the cosine formula, especially within the context of its paired identity with the sine formula, enhances the ability to solve various trigonometric problems effectively.

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, tangent, and others. Solving these equations requires using identities and simplifying strategies.
A key to working with trigonometric equations is recognizing and applying identities like the sine and cosine formulas, as well as other identities like the Pythagorean identities.
In the given exercise, a balanced equation with trigonometric components was simplified. The goal often is to manipulate the equation by substituting equivalent expressions or rearranging terms.
For example, converting an equation using the addition formulas helps simplify complex expressions. The addition formula states:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

By using these, much like in the original exercise, we simplify trigonometric expressions to show equivalence or identity between two sides of the equation.
Approaching trigonometric equations methodically, using known identities to transform and solve expressions, is essential for proving statements or solving problems efficiently.