If `theta in [0,5pi]` and `r in R` such that `2sintheta=r^4-2r^2+3` then the maximum number of values of the pair `(r,theta)` is.....

Explanation

To solve the problem, we need to analyze the equation given:

\[ 2 \sin \theta = r^4 - 2r^2 + 3 \]

### Step 1: Understand the range of \( 2 \sin \theta \)

The sine function, \( \sin \theta \), varies between -1 and 1. Therefore, \( 2 \sin \theta \) will vary between:

...