The minimum value of `2 sin^2 theta + 3cos^2 theta` is

Options

A. 0
B. 3
C. 2
D. 1

Answer

To find the minimum value of the expression \(2 \sin^2 \theta + 3 \cos^2 \theta\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 2 \sin^2 \theta + 3 \cos^2 \theta \]
We can rewrite this as:
\[ 2 \sin^2 \theta + 3 (1 - \sin^2 \theta) = 2 \sin^2 \theta + 3 - 3 \sin^2 \theta \]
This simplifies to:
\[ (2 - 3) \sin^2 \theta + 3 = -\sin^2 \theta + 3 \]

### Step 2: Analyze the new expression
Now we have:
\[ -\sin^2 \theta + 3 \]
To find the minimum value of this expression, we need to consider the range of \(\sin^2 \theta\).

### Step 3: Determine the range of \(\sin^2 \theta\)
The value of \(\sin^2 \theta\) varies between 0 and 1:
\[ 0 \leq \sin^2 \theta \leq 1 \]

### Step 4: Substitute the maximum value of \(\sin^2 \theta\)
To find the minimum value of \(-\sin^2 \theta + 3\), we substitute the maximum value of \(\sin^2 \theta\):
\[ -\sin^2 \theta + 3 \text{ is minimized when } \sin^2 \theta \text{ is maximized.} \]
Thus, substituting \(\sin^2 \theta = 1\):
\[ -\sin^2 \theta + 3 = -1 + 3 = 2 \]

### Step 5: Conclusion
Therefore, the minimum value of \(2 \sin^2 \theta + 3 \cos^2 \theta\) is:
\[ \boxed{2} \]

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