What is the least value of 15 `cos^2 theta + 17 sin^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 can express the given function in terms of \( \sin^2 \theta \) and \( \cos^2 \theta \):
\[ f(\theta) = 15 \cos^2 \theta + 17 \sin^2 \theta \]

### Step 2: Use the Identity
Recall the Pythagorean identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
We can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting this into the function gives:
\[ f(\theta) = 15(1 - \sin^2 \theta) + 17 \sin^2 \theta \]
\[ = 15 - 15 \sin^2 \theta + 17 \sin^2 \theta \]
\[ = 15 + 2 \sin^2 \theta \]

### Step 3: Analyze the Function
The term \( 2 \sin^2 \theta \) varies between 0 and 2, since \( \sin^2 \theta \) ranges from 0 to 1. Therefore:
\[ f(\theta) = 15 + 2 \sin^2 \theta \]
will range from:
\[ 15 + 2(0) = 15 \quad \text{to} \quad 15 + 2(1) = 17 \]

### Step 4: Determine the Minimum Value
The least value of \( f(\theta) \) occurs when \( \sin^2 \theta = 0 \):
\[ \text{Minimum value} = 15 + 2(0) = 15 \]

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