Find the general solution of `1+sin^3x + cos^(3)x=(3)/(2)sin 2x `

Answer

To find the general solution of the equation \(1 + \sin^3 x + \cos^3 x = \frac{3}{2} \sin 2x\), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ 1 + \sin^3 x + \cos^3 x = \frac{3}{2} \sin 2x \]

### Step 2: Use the Identity for \(\sin 2x\)
Recall that \(\sin 2x = 2 \sin x \cos x\). We can rewrite the right-hand side:
\[ 1 + \sin^3 x + \cos^3 x = 3 \sin x \cos x \]

### Step 3: Apply the Sum of Cubes Formula
Using the identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), we can express \(\sin^3 x + \cos^3 x\):
Let \(a = \sin x\) and \(b = \cos x\):
\[ \sin^3 x + \cos^3 x = (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x) \]
Since \(\sin^2 x + \cos^2 x = 1\), we have:
\[ \sin^3 x + \cos^3 x = (\sin x + \cos x)(1 - \sin x \cos x) \]

### Step 4: Substitute Back into the Equation
Substituting this back into our equation gives:
\[ 1 + (\sin x + \cos x)(1 - \sin x \cos x) = 3 \sin x \cos x \]

### Step 5: Rearranging the Equation
Rearranging the equation, we have:
\[ 1 + \sin x + \cos x - \sin x \cos x (\sin x + \cos x) = 3 \sin x \cos x \]
This simplifies to:
\[ 1 + \sin x + \cos x = 3 \sin x \cos x + \sin x \cos x (\sin x + \cos x) \]
Factoring out \(\sin x \cos x\) on the right side:
\[ 1 + \sin x + \cos x = \sin x \cos x (3 + \sin x + \cos x) \]

### Step 6: Setting Up the Equation
Now we can set up the equation:
\[ 1 + \sin x + \cos x = \sin x \cos x (3 + \sin x + \cos x) \]

### Step 7: Solving for \(\sin x + \cos x\)
Let \(y = \sin x + \cos x\). Then, we can express \(\sin x \cos x\) in terms of \(y\):
\[ \sin x \cos x = \frac{y^2 - 1}{2} \]
Substituting this back into our equation gives:
\[ 1 + y = \frac{y^2 - 1}{2} (3 + y) \]

### Step 8: Simplifying the Equation
Multiply through by 2 to eliminate the fraction:
\[ 2(1 + y) = (y^2 - 1)(3 + y) \]
Expanding both sides:
\[ 2 + 2y = 3y^2 + y^3 - 3 - y \]
Rearranging gives:
\[ y^3 + 3y^2 - 3y - 5 = 0 \]

### Step 9: Finding Roots of the Cubic Equation
We can use the Rational Root Theorem or synthetic division to find the roots of this cubic equation. After finding the roots, we can express \(y\) in terms of \(x\).

### Step 10: General Solution
Once we have the values of \(y\), we can find \(x\) using:
\[ \sin x + \cos x = y \]
This leads to:
\[ \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = y \]
From here, we can derive the general solutions for \(x\).