The least value of `3sin^2theta+4cos^2theta` is:

Options

A. 1
B. 2
C. 3
D. 5

Answer

To find the least value of the expression \(3\sin^2\theta + 4\cos^2\theta\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 3\sin^2\theta + 4\cos^2\theta \]
We can rewrite \(4\cos^2\theta\) as \(3\cos^2\theta + \cos^2\theta\):
\[ 3\sin^2\theta + 3\cos^2\theta + \cos^2\theta \]

### Step 2: Factor out the common term
Now, we can factor out the common term \(3\) from the first two terms:
\[ 3(\sin^2\theta + \cos^2\theta) + \cos^2\theta \]

### Step 3: Use the Pythagorean identity
Using the Pythagorean identity, we know that:
\[ \sin^2\theta + \cos^2\theta = 1 \]
Substituting this into our expression gives:
\[ 3(1) + \cos^2\theta = 3 + \cos^2\theta \]

### Step 4: Determine the least value of \(\cos^2\theta\)
The term \(\cos^2\theta\) can take values from \(0\) to \(1\) since it is a square of the cosine function. The least value occurs when \(\cos^2\theta = 0\).

### Step 5: Calculate the least value of the expression
Substituting the least value of \(\cos^2\theta\) into the expression:
\[ 3 + 0 = 3 \]

### Conclusion
Thus, the least value of \(3\sin^2\theta + 4\cos^2\theta\) is:
\[ \boxed{3} \]