Number of values of `p` for which equation `sin^3x+1+p^3-3psinx=0(p >0) has a root is ________

Explanation

To solve the equation \( \sin^3 x + 1 + p^3 - 3p \sin x = 0 \) for the number of values of \( p \) such that the equation has a root, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \sin^3 x + 1 + p^3 - 3p \sin x = 0 \]
Rearranging gives us:
\[ \sin^3 x + p^3 = 3p \sin x - 1 \]

### Step 2: Substituting \( y = \sin x \)
Let \( y = \sin x \). The equation becomes:
\[ y^3 + p^3 = 3py - 1 \]
This can be rewritten as:
\[ y^3 - 3py + (p^3 + 1) = 0 \]

### Step 3: Analyzing the Cubic Equation
We need to analyze the cubic equation:
\[ y^3 - 3py + (p^3 + 1) = 0 \]
This is a cubic equation in \( y \). For this cubic equation to have at least one real root, we can use the discriminant or analyze the behavior of the function.

### Step 4: Finding the Critical Points
To find the critical points, we differentiate the function:
\[ f(y) = y^3 - 3py + (p^3 + 1) \]
The derivative is:
\[ f'(y) = 3y^2 - 3p \]
Setting the derivative to zero gives:
\[ 3y^2 - 3p = 0 \implies y^2 = p \implies y = \pm \sqrt{p} \]

### Step 5: Evaluating the Function at Critical Points
Now we evaluate the function \( f(y) \) at the critical points \( y = \sqrt{p} \) and \( y = -\sqrt{p} \):
1. For \( y = \sqrt{p} \):
\[ f(\sqrt{p}) = (\sqrt{p})^3 - 3p(\sqrt{p}) + (p^3 + 1) = p\sqrt{p} - 3p\sqrt{p} + (p^3 + 1) = -2p\sqrt{p} + (p^3 + 1) \]
2. For \( y = -\sqrt{p} \):
\[ f(-\sqrt{p}) = (-\sqrt{p})^3 - 3p(-\sqrt{p}) + (p^3 + 1) = -p\sqrt{p} + 3p\sqrt{p} + (p^3 + 1) = 2p\sqrt{p} + (p^3 + 1) \]

### Step 6: Conditions for Roots
For the cubic equation to have at least one real root, one of these evaluations must be zero or change signs.

1. Set \( f(\sqrt{p}) = 0 \):
\[ -2p\sqrt{p} + (p^3 + 1) = 0 \implies p^3 + 1 = 2p\sqrt{p} \]

2. Set \( f(-\sqrt{p}) = 0 \):
\[ 2p\sqrt{p} + (p^3 + 1) = 0 \implies p^3 + 1 = -2p\sqrt{p} \]

### Step 7: Finding the Values of \( p \)
We need to find the values of \( p \) such that these conditions hold true. After solving these equations, we find that the only value of \( p \) that satisfies the conditions and is greater than zero is \( p = 1 \).

### Conclusion
Thus, the number of values of \( p \) for which the equation has a root is:
\[ \boxed{1} \]