Find all the numbers ‘a’ for which any root of the equation

Sin 3x = a sin x + (4-2|a|) sin² x is a root of the equation

Sin 3x + Cos 2x = 1 + 2 Sinx Cos2x and any root of the latter equation is a root of the former.

To tackle the problem of finding all the values of 'a' for which the roots of the equations Sin 3x = a sin x + (4 - 2|a|) sin 2x and Sin 3x + Cos 2x = 1 + 2 Sin x Cos 2x are interchangeable, we need to analyze both equations carefully. Let's break this down step by step.

Understanding the First Equation

The first equation we have is:

Sin 3x = a sin x + (4 - 2|a|) sin 2x

This equation can be rewritten using the triple angle formula for sine:

Sin 3x = 3 sin x - 4 sin^3 x

Thus, we can equate:

3 sin x - 4 sin^3 x = a sin x + (4 - 2|a|) sin 2x

Analyzing the Second Equation

The second equation is:

Sin 3x + Cos 2x = 1 + 2 Sin x Cos 2x

Using the identity Cos 2x = 1 - 2 sin^2 x, we can rewrite this as:

Sin 3x + (1 - 2 sin^2 x) = 1 + 2 Sin x (1 - 2 sin^2 x)

Simplifying this gives:

Sin 3x - 2 sin^2 x = 2 Sin x - 4 Sin^3 x

Finding Common Roots

For any root of the first equation to also be a root of the second, and vice versa, we need to ensure that the two equations are equivalent under certain conditions. This means that the coefficients of corresponding terms must match.

Setting Up the Coefficients

From the first equation, we can express it as:

• Coefficient of sin x: 3 - a
• Coefficient of sin^2 x: 0
• Coefficient of sin^3 x: -4 + 2|a|

From the second equation, we have:

• Coefficient of sin x: 2
• Coefficient of sin^2 x: -2
• Coefficient of sin^3 x: -4

Equating Coefficients

Now, we can set up the following equations based on the coefficients:

• 3 - a = 2
• 0 = -2
• -4 + 2|a| = -4

Solving the Equations

From the first equation, we can solve for 'a':

3 - a = 2 ⇒ a = 1

The second equation, 0 = -2, is not valid, which indicates that there are no restrictions from this coefficient. The third equation simplifies to:

2|a| = 0 ⇒ |a| = 0 ⇒ a = 0

Final Values of 'a'

Thus, the values of 'a' that satisfy the conditions for the roots of both equations to be interchangeable are:

• a = 1
• a = 0

In summary, the values of 'a' for which any root of the first equation is a root of the second equation, and vice versa, are 0 and 1. This means that both equations will share common roots under these conditions, allowing for a deeper exploration of their behavior in trigonometric contexts.