Minimum value of `3 sin theta+4 cos theta` in the interval `[0, (pi)/(2)]` is : (a) `-5` (b) `3` (c) `4` (d) `(7)/(sqrt(2))`

Options

A. -5
B. 3
C. 4
D. `(7)/(sqrt(2))`

Answer

To find the minimum value of the expression \(3 \sin \theta + 4 \cos \theta\) in the interval \([0, \frac{\pi}{2}]\), we can follow these steps:

### Step 1: Identify the coefficients
Let \(a = 3\) and \(b = 4\). We will use these values to find the minimum and maximum of the expression.

### Step 2: Calculate the magnitude
We need to calculate \(\sqrt{a^2 + b^2}\):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Determine the minimum value
The minimum value of the expression \(a \sin \theta + b \cos \theta\) is given by:
\[ -\sqrt{a^2 + b^2} \]
Thus, the minimum value is:
\[ -\sqrt{25} = -5 \]

### Step 4: Conclusion
The minimum value of \(3 \sin \theta + 4 \cos \theta\) in the interval \([0, \frac{\pi}{2}]\) is \(-5\).

### Final Answer
The correct option is (a) \(-5\).

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