Find the minimum value of 2 sin 8 + 3 cos 8

Answer

To find the minimum value of the expression \(2 \sin 8 + 3 \cos 8\), we can use the method of combining sine and cosine into a single sinusoidal function.

### Step-by-Step Solution:

1. **Identify the coefficients**:
We have \(a = 2\) (coefficient of \(\sin 8\)) and \(b = 3\) (coefficient of \(\cos 8\)).

2. **Calculate \(a^2 + b^2\)**:
\[ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \]

3. **Calculate the maximum and minimum values**:
- The maximum value of \(2 \sin 8 + 3 \cos 8\) is given by \(\sqrt{a^2 + b^2}\):
\[ \text{Maximum} = \sqrt{13} \]
- The minimum value is given by \(-\sqrt{a^2 + b^2}\):
\[ \text{Minimum} = -\sqrt{13} \]

4. **Final Result**:
Therefore, the minimum value of \(2 \sin 8 + 3 \cos 8\) is:
\[ -\sqrt{13} \]