If ` theta in [0,3 pi] and r in R ` . Then , find the pair of `( r , theta )` satisfying` 2 sin theta = r^(4)- 2r^(2)+3 ` .

Answer

To solve the equation \( 2 \sin \theta = r^4 - 2r^2 + 3 \) for \( \theta \in [0, 3\pi] \) and \( r \in \mathbb{R} \), we will follow these steps:

### Step 1: Analyze the range of \( 2 \sin \theta \)
The sine function has a range of \([-1, 1]\). Therefore, the range of \( 2 \sin \theta \) is:
\[ [-2, 2] \]

### Step 2: Rewrite the equation
We can rewrite the equation \( r^4 - 2r^2 + 3 \) as:
\[ r^4 - 2r^2 + 3 = (r^2 - 1)^2 + 2 \]
This shows that the expression \( r^4 - 2r^2 + 3 \) is always greater than or equal to 2 because the square term \((r^2 - 1)^2\) is non-negative.

### Step 3: Set up the inequality
Since \( 2 \sin \theta \) can take values from \(-2\) to \(2\), we need to find when:
\[ (r^2 - 1)^2 + 2 \leq 2 \]
This simplifies to:
\[ (r^2 - 1)^2 \leq 0 \]
The only solution to this inequality is when:
\[ (r^2 - 1)^2 = 0 \implies r^2 - 1 = 0 \implies r^2 = 1 \implies r = \pm 1 \]

### Step 4: Find values of \( \theta \)
Now, we substitute \( 2 \sin \theta = 2 \) (the maximum value) to find \( \theta \):
\[ \sin \theta = 1 \]
The general solutions for \( \sin \theta = 1 \) are:
\[ \theta = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]
Within the interval \( [0, 3\pi] \), the valid values of \( \theta \) are:
- For \( n = 0 \): \( \theta = \frac{\pi}{2} \)
- For \( n = 1 \): \( \theta = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} \)
- For \( n = 2 \): \( \theta = \frac{\pi}{2} + 4\pi = \frac{9\pi}{2} \) (not in the range)

Thus, the valid values of \( \theta \) are \( \frac{\pi}{2} \) and \( \frac{5\pi}{2} \).

### Step 5: List pairs of \( (r, \theta) \)
We have two values for \( r \) (\( r = 1 \) and \( r = -1 \)) and two valid values for \( \theta \) (\( \frac{\pi}{2} \) and \( \frac{5\pi}{2} \)). Therefore, the pairs \( (r, \theta) \) are:
1. \( (1, \frac{\pi}{2}) \)
2. \( (1, \frac{5\pi}{2}) \)
3. \( (-1, \frac{\pi}{2}) \)
4. \( (-1, \frac{5\pi}{2}) \)

### Conclusion
The total number of pairs \( (r, \theta) \) satisfying the equation is \( 4 \).