The minimum value of `a sin theta+b cos theta` is:

Options

A. `sqrt(a^(2)+b^(2))`
B. `-sqrt(a^(2)-b^(2))`
C. `-sqrt(a^(2)+b^(2))`
D. None of these

Answer

To find the minimum value of the expression \( a \sin \theta + b \cos \theta \), we can use the following steps:

### Step 1: Identify the expression
The expression we are dealing with is:
\[ f(\theta) = a \sin \theta + b \cos \theta \]

### Step 2: Use the formula for maximum and minimum values
The maximum and minimum values of the expression \( a \sin \theta + b \cos \theta \) can be derived using the formula:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} \]
\[ \text{Minimum value} = -\sqrt{a^2 + b^2} \]

### Step 3: Apply the formula
For our expression \( f(\theta) = a \sin \theta + b \cos \theta \):
- The minimum value is:
\[ \text{Minimum value} = -\sqrt{a^2 + b^2} \]

### Step 4: Conclusion
Thus, the minimum value of \( a \sin \theta + b \cos \theta \) is:
\[ -\sqrt{a^2 + b^2} \]

### Final Answer:
The minimum value of \( a \sin \theta + b \cos \theta \) is \( -\sqrt{a^2 + b^2} \).

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