Question

Largest value of \( \cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2 \)

• A. 4
• B. 0
• C. \( 4 + \sqrt{10} \)
• D. \( 4 - \sqrt{10} \)

Hint:

To maximize expressions involving trigonometric functions, look for symmetry and consider completing the square or using calculus for more complex forms.

Answer 1

The Correct Option is C

We are given the expression: \[ f(\theta) = \cos^2 \theta - 6 \sin \theta \cos \theta + 3 \sin^2 \theta + 2 \] Step 1: Simplifying the Expression First, notice that \( \cos^2 \theta + \sin^2 \theta = 1 \). We can try to express the equation in terms of either \( \sin \theta \) or \( \cos \theta \), but we will begin by simplifying the trigonometric identity. Let \( x = \cos \theta \) and \( y = \sin \theta \). We know that: \[ x^2 + y^2 = 1 \] So, the expression becomes: \[ f(\theta) = x^2 - 6xy + 3y^2 + 2 \] Step 2: Maximize the Expression To find the maximum value, we can take the derivative of \( f(\theta) \) with respect to \( \theta \). However, it's easier to notice that the quadratic terms in \( x \) and \( y \) suggest completing the square. After completing the square, we find that the maximum value occurs when \( x = \cos \theta \) and \( y = \sin \theta \). Through algebraic simplification and maximizing, we get the largest value of the expression as: \[ \boxed{4 + \sqrt{10}} \] Thus, the correct answer is \( \boxed{(c) \, 4 + \sqrt{10}} \).