Problem 3 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a tringle \(\Delta A B C\), prove that \(\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (C-A)}{\sin C}=0\)

Step-by-Step Solution

Answer

The equation \(\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (A-B)}{\sin C}=0\) is proved by replacing \(\sin A\), \(\sin B\), and \(\sin C\) with the sine rule in a triangle, and then using the property that sum of sines of angles of form (A-B), (B-C), and (C-A) equals 0.

Step 1: Begin the proof

Start by writing the given equation that needs to be proved, which is \(\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (A-B)}{\sin C}=0\).

Step 2: Apply the sine rule

Notice that \(\sin A\), \(\sin B\), and \(\sin C\) can be replaced with \(\frac{a}{2R}\), \(\frac{b}{2R}\), and \(\frac{c}{2R}\) respectively from the sine rule in the triangle. This would give: \[(2R)^{2}\left(\frac{a \sin (B-C)}{a}+\frac{b \sin (C-A)}{b}+\frac{c \sin (A-B)}{c}\right)=0\].

Step 3: Simplify the equation

The equation can now be simplified by cancelling out similar terms to get: \[(2R)^{2}\left(\sin (B-C)+\sin (C-A)+\sin (A-B)\right)=0\].

Step 4: Use the property of sine function

Sum of sines of angles of form (A-B), (B-C), and (C-A) is known to equal 0. Hence, \[(2R)^{2} \cdot 0 = 0\].

Step 5: Final verification

The equation is now proved, as the right side of the equation is indeed 0.

Key Concepts

Sine Rule

The sine rule is a fundamental relationship in trigonometry that applies to any triangle, not just right-angled ones. It links the sides of a triangle to the sines of its angles, and is expressed as follows:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \)

Here, \(a\), \(b\), and \(c\) are the lengths of the sides opposite angles \(A\), \(B\), and \(C\) respectively, while \(R\) is the circumradius of the triangle, meaning the radius of the circle in which the triangle is inscribed.
Using the sine rule, you can solve for unknown sides or angles when given enough other information. It's especially helpful in problems involving obtuse or acute triangles.
This rule simplifies the manipulation of trigonometric problems, allowing the replacement of sine angles in complex equations with easy-to-handle side ratios.
In the original exercise, the sine rule helps express \/(\sin A \/), \/(\sin B \/), and \/(\sin C \/) in terms of side length \(a, b, c\) and the circumradius \(2R\), clarifying the path to solve the equation.

Triangular Identities

Triangular identities are intrinsic properties and relationships found within the angles and sides of triangles. They help in transforming complex trigonometric expressions to simpler forms.
Two of the most common identities are the angle sum identity \( \sin(A + B + C) = 0 \) for any triangle and the angle subtraction formulas.

• In any triangle, \( A + B + C = 180^\circ \) or \( \pi \) radians.
• These identities are often used in conjunction with laws like the sine rule to prove more complex relationships.

For example, the original exercise leverages the property where the sum of the sine of angles \( (B-C), (C-A), \) and \( (A-B) \) is zero, simplifying the equation to conclude it equals zero. This property is particularly useful in proving symmetrical equations like the one given in the exercise.
Understanding these identities provides a stronger grasp of the relationships within triangles, offering a toolkit for solving many geometric and trigonometric problems.

Angle Subtraction Identity

The angle subtraction identity is another useful tool in trigonometry, equipping students to manage expressions where multiple angles interact. It is articulated for sine as:

• \( \sin(X - Y) = \sin X \cos Y - \cos X \sin Y \)

This identity helps simplify expressions that involve the differences between angles, making it an essential component in proofs dealing with angle differences.
In the context of this exercise, the angle subtraction identities translate terms like \( \sin(B-C) \), \( \sin(C-A) \), and \( \sin(A-B) \) into manageable forms. By breaking down these terms, this identity assists in cancelling and simplifying relation-heavy terms that stem from angular subtractions.
Employing angle subtraction identities, alongside the sine rule and triangular identities, provides a structured approach to breaking down and tackling traditionally complex trigonometric proofs efficiently and accurately.