Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x \sin a+y \sin 2 a+z \sin 3 a=\sin 4 a x \sin b+\) \(y \sin 2 b+z \sin 3 b=\sin 4 b x \sin c+y \sin 2 c+z\) \(\sin 3 c=\sin 4 c\), then the roots of the equation \(t^{3}-\frac{z}{2} t^{2}-\frac{y+2}{4}+\frac{z-x}{8}=0 ; a, b, c \neq n \pi\), are (A) \(\sin a, \sin b, \sin c\) (B) \(\cos a, \cos b, \cos c\) (C) \(\sin 2 a, \sin 2 b, \sin 2 c\) (D) \(\cos 2 a, \cos 2 b, \cos 2 c\)

Step-by-Step Solution

Answer

The roots are \(\sin a, \sin b, \sin c\) (Option A).

Step 1: Identify Values for Sines

Given the equations, we need to deduce the relationships: 1. We have that each \(z \sin 3a\), \(z \sin 3b\), and \(z \sin 3c \) match with \(\sin 4a\), \(\sin 4b\), and \(\sin 4c\) respectively, these equations imply that \(\sin 3a = \sin 4a\), \(\sin 3b = \sin 4b\), and \(\sin 3c = \sin 4c\). This sets an identity that must hold: \(\sin \theta = \sin (2\pi - \theta)\) given the periodic property of sine.

Step 2: Transformations with Trigonometric Identities

Each sine equation can be aligned via transformations:We note \(\sin 3\theta - \sin 4\theta = 0\) simplifying to \(\sin\theta = \sin\phi\). This suggests \(\theta = \phi + k\pi \) (for integer \(k\)), and consistent angle transformations imply possible simplifications to: \(\theta = a, b, c\).

Step 3: Evaluate Roots in Original Polynomial

The target polynomial is provided as \[t^3 - \frac{z}{2} t^2 - \frac{y+2}{4} t + \frac{z-x}{8} = 0\].Comparing to the transformations from step 2, roots \(\sin a, \sin b, \sin c\) agree with these identity transitions, imposes trigonometric similarities among the angles of possible solutions in regard to inherent transformations for sine.

Step 4: Confirm Possible Solutions

Evaluate if sine values appear within logical constraints:Given the provided sine equational matches, solutions are considered valid across \(sin n \pi\), particularly for \(t = \sin a, \sin b, \sin c)\). Thus roots match options in form \sin n a , ..., \sin n c\.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is key to unraveling the given problem. Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable where both sides of the equation are defined. One important identity among them is the periodicity of sine:

• \( \sin(\theta) = \sin(2\pi - \theta) \)

This essentially means the sine function repeats its values every \(2\pi\) radians, or 360 degrees.

In the exercise, we use this principle to transition through transformations like \( \sin 3\theta - \sin 4\theta = 0 \), simplifying to \( \sin \theta = \sin \phi \). Simplifying a trigonometric expression usually involves setting angles like \( \theta \) or \( \phi \) to be offset by multiples of \( \pi \), such as \( \theta = \phi + k\pi \), ensuring that we encompass all possible angle values. This understanding allows us to form equivalences that make expressions easier to handle, as seen when looking for polynomial roots.

Polynomial Roots

The problem involves determining the roots of a cubic polynomial:

• \[ t^3 - \frac{z}{2} t^2 - \frac{y+2}{4} t + \frac{z-x}{8} = 0 \]

Roots of a polynomial are the values of \(t\) for which the polynomial equals zero.

Finding these roots involves factoring the polynomial, using methods such as the Rational Root Theorem, or synthetic division if applicable. In the context of the problem, connections between the expression given in trigonometric terms and the roots of the polynomial are established by examining trigonometric identities and substituting known sine values. Because the sine values have particular relationships as dictated by step 1, these reciprocate in the appearance of roots like \( \sin a, \sin b, \sin c \), aligning to certain functional properties of \(t\) inside the polynomial.

Sine and Cosine Values

Sine and cosine are fundamental to trigonometry, defining the primary relationships in right triangles. In the exercise, we focus mainly on sine values but the cosine is equally essential.

The sine \( \sin(\theta) \) is the ratio of the length of the side opposite \( \theta \) to the hypotenuse in a right triangle. These values oscillate between -1 and 1 as \( \theta \) varies. Similarly, cosine \( \cos(\theta) \) is the ratio of the adjacent side to the hypotenuse. Knowing and leveraging these values helps in solving equations like those provided, especially when exploring equations and identities like \( \sin 3a = \sin 4a \).

• Sine values repeat every \( 2\pi \) radians.
• Key angles include 0, \( \pi/2 \), \( \pi \), and \( 3\pi/2 \).

In the problem, substituting sine values drawn from these consistent properties permits easy transition to understanding their effect within polynomial equations, aiding greatly in discovering potential roots like \( \sin a, \sin b, \sin c \).