Problem 287 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \sin x+\sin y=3(\cos y-\cos x), \text { prove that } \sin 3 x+\sin 3 y=0 $$

Step-by-Step Solution

Answer

Proved that \(\sin 3 x + \sin 3 y = 0\) holds true for the given equation \(\sin x + \sin y = 3 (\cos y - \cos x)\).

Step 1: Prepare the Expression

The given equation is \(\sin x + \sin y = 3(\cos y - \cos x)\), rewrite it into \(\sin x + \sin y - 3(\cos y - \cos x) = 0\).

Step 2: Apply the Trigonometric Identities

We apply the identity \(\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos\frac{A-B}{2}\) on the left side of the equation. It becomes: \(2 \sin \frac{x+y}{2} \cos\frac{x-y}{2}\). \nWe apply the identity \(\cos A - \cos B = -2 \sin\frac{A+B}{2} \sin\frac{A-B}{2}\) on the right side of the equation. It becomes: \(3 * 2 \sin\frac{x+y}{2} \sin\frac{x-y}{2}\). \nThe whole equation becomes: \(2 \sin \frac{x+y}{2} \cos\frac{x-y}{2} - 3 * 2 \sin\frac{x+y}{2} \sin\frac{x-y}{2} = 0\).

Step 3: Simplify the Equation

After simplification, since both sides share the same \(\sin\) function, we could condense the equation to \(\cos\frac{x-y}{2} - 3\sin\frac{x-y}{2} = 0\).

Step 4: Apply the Trigonometric Identity

Applying the identity \(\sin 3 A = 3 \sin A - 4 \sin^3 A\) on \(\sin\frac{x-y}{2}\), we get \(\sin 3(\frac{x-y}{2}) = 0\), which indicates \(\sin 3x = \sin 3y\). From the rules of sine function, \(\sin A = \sin B\) means \(A = B + n\pi\), where \(n\) is an integer. Therefore, we have \(3x = 3y + n\pi\). When \(n\) equals 0, \(x = y\), then \(\sin 3x + \sin 3y = 2 \sin 3x = 0\). Also when \(n\) equals 1, \(\sin 3x + \sin 3(y + \pi) = 0\). Hence, we proved that \(\sin 3 x + \sin 3 y = 0\) indeed holds true.

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental to trigonometry. They are periodic functions, meaning they repeat their values in regular intervals. Sine and cosine are especially known for their application in right-angle triangles and the unit circle.

• **Sine Function, \( \sin \theta \):** Represents the ratio of the length of the side opposite to the angle \( \theta \) to the hypotenuse in a right triangle. On the unit circle, it corresponds to the y-coordinate of a point.
• **Cosine Function, \( \cos \theta \):** Represents the ratio of the length of the adjacent side to the angle \( \theta \) to the hypotenuse in a right triangle. On the unit circle, it corresponds to the x-coordinate of a point.

In any right triangle, the Pythagorean identity holds: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is foundational because it symbolizes the relationship between these two functions at any angle \( \theta \).By understanding these functions, you can easily navigate through trigonometric problems, including solving equations or simplifying expressions using identities.

Angle Sum and Difference Formulas

Angle Sum and Difference Formulas help calculate the sine, cosine, or tangent of two combined angles. They are invaluable when dealing with trigonometric problems and help derive other popular identities.

• **Sum Formulas:**
  • \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
  • \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• **Difference Formulas:**
  • \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
  • \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These formulas are often used to express complex trigonometric equations in a simpler form, making calculations more manageable.
For instance, when asked to express \( \sin x + \sin y \) or \( \cos y - \cos x \) as seen in the exercise, these kinds of transformations are extremely beneficial.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions, which require solutions within certain ranges. They can range from simple to complex but can often be solved using trigonometric identities or transformations.

• **Solving Process:** Often involves the transformation of the given equation into a simpler form using identities and simplifications.
• **Key Identities:** Help relate different trigonometric functions to each other, such as \( \sin A = \sin B \), implying \( A = B + n\pi \), where \( n \) is an integer.
• **Periodicity:** Trigonometric functions repeat their values at specific intervals, which means solutions can often have multiple instances.

In the given problem, you can observe how transformations and identities are used to prove that \( \sin 3x + \sin 3y = 0 \). By condensing the equation and applying identities, you derive possible values of \( x \) and \( y \) satisfy the initial equation. It illustrates the powerful role that understanding trigonometric identities can play in solving such equations.