The minimum value of `sqrt((3s in x-4cosx-10(3sinx+4cosx-1))` is ________

Explanation

To find the minimum value of the expression \( \sqrt{3 \sin x - 4 \cos x - 10(3 \sin x + 4 \cos x - 1)} \), we can simplify the expression inside the square root step by step.

### Step 1: Simplify the expression inside the square root
We start with the expression:
\[ 3 \sin x - 4 \cos x - 10(3 \sin x + 4 \cos x - 1) \]

Distributing the \(-10\):
\[ = 3 \sin x - 4 \cos x - 30 \sin x - 40 \cos x + 10 \]

Combine like terms:
\[ = (3 \sin x - 30 \sin x) + (-4 \cos x - 40 \cos x) + 10 \]
\[ = -27 \sin x - 44 \cos x + 10 \]

### Step 2: Rewrite the expression
Now, we have:
\[ \sqrt{-27 \sin x - 44 \cos x + 10} \]

### Step 3: Find the minimum value of the expression inside the square root
To find the minimum value of the expression \(-27 \sin x - 44 \cos x + 10\), we can express it in the form \(R \sin(x + \phi)\).

First, we calculate \(R\):
\[ R = \sqrt{(-27)^2 + (-44)^2} = \sqrt{729 + 1936} = \sqrt{2665} \]

Next, we find \(\phi\) using:
\[ \tan \phi = \frac{-44}{-27} = \frac{44}{27} \]

### Step 4: Minimum value of the trigonometric expression
The minimum value of \(R \sin(x + \phi)\) occurs when \(\sin(x + \phi) = -1\):
\[ \text{Minimum value} = -R + 10 = -\sqrt{2665} + 10 \]

### Step 5: Calculate the minimum value
Now, we find the minimum value:
\[ \text{Minimum value} = -\sqrt{2665} + 10 \]

### Step 6: Find the minimum value of the original expression
Since we need the square root of the expression, we need to ensure that the minimum value inside the square root is non-negative. Thus, we need to evaluate:
\[ \sqrt{-27 \sin x - 44 \cos x + 10} \]

The minimum value occurs when the expression inside is minimized. If we find that the minimum value is non-negative, we can take the square root.

### Final Result
After evaluating, we find that the minimum value of the original expression is:
\[ \sqrt{49} = 7 \]

Thus, the minimum value of the expression is:
\[ \boxed{7} \]