What is the least value of `15cos^(2)theta+17sin^(2)theta`?

Options

A. 14
B. 15
C. 2
D. 18

Answer

To find the least value of the expression \( 15\cos^2\theta + 17\sin^2\theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 15\cos^2\theta + 17\sin^2\theta \]

### Step 2: Factor out common terms
We can rewrite the expression by factoring out \( 15 \):
\[ 15\cos^2\theta + 15\sin^2\theta + 2\sin^2\theta = 15(\cos^2\theta + \sin^2\theta) + 2\sin^2\theta \]

### Step 3: Use the Pythagorean identity
Using the Pythagorean identity \( \cos^2\theta + \sin^2\theta = 1 \):
\[ 15(1) + 2\sin^2\theta = 15 + 2\sin^2\theta \]

### Step 4: Analyze the term \( 2\sin^2\theta \)
The term \( \sin^2\theta \) varies between 0 and 1. Therefore, \( 2\sin^2\theta \) will vary between:
\[ 2 \cdot 0 = 0 \quad \text{and} \quad 2 \cdot 1 = 2 \]

### Step 5: Find the least value
The least value of \( 2\sin^2\theta \) is 0, which occurs when \( \sin^2\theta = 0 \). Thus, substituting this back into our expression gives:
\[ 15 + 0 = 15 \]

### Conclusion
Therefore, the least value of \( 15\cos^2\theta + 17\sin^2\theta \) is:
\[ \boxed{15} \]

---