Question

The minimum value of \( \cos^2\theta + 6\sin\theta\cos\theta + 3\sin^2\theta \) is:

• A. $-1$
• B. $1$
• C. $5 + \sqrt{10}$
• D. $5 - \sqrt{10}$

Hint:

Expressions of the form \(a\sin^2\theta + b\sin\theta\cos\theta + c\cos^2\theta\) can be minimized using eigenvalue or standard minimum formulas.

Answer 1

The Correct Option is D

1. To find the minimum value of the expression \( \cos^2\theta + 6\sin\theta\cos\theta + 3\sin^2\theta \), we start by recognizing a pattern or form that can simplify our calculations.
2. First, recall the identities:
   • \( \cos^2\theta + \sin^2\theta = 1 \)
   • Let \( \cos\theta = x \) and \( \sin\theta = y \). This makes \( x^2 + y^2 = 1 \).
3. Substituting these identities, the expression becomes: \(x^2 + 6xy + 3y^2\)
4. Since \( x^2 + y^2 = 1 \), we have \( x^2 = 1 - y^2 \). Substitute this in: \((1-y^2) + 6xy + 3y^2 = 1 + 6xy + 2y^2\)\)
5. We aim to minimize \( 1 + 6xy + 2y^2 \). Notice this is a quadratic in terms of \( y \), so let's rewrite it: \(2y^2 + 6xy + 1\)
6. Use the method of completing the square on the quadratic expression. Rewrite as: \(2\left(y^2 + \frac{3x}{2}y\right) + 1\)
7. The completing square step involves taking half of the coefficient of \( y \), squaring it, and adjusting the constant term: - Half of \(\frac{3x}{2}\) is \(\frac{3x}{4}\). - Square of \(\frac{3x}{4}\) is \(\frac{9x^2}{16}\).
8. Re-write: \(2\left((y + \frac{3x}{4})^2 - \frac{9x^2}{16}\right) + 1\)\)
9. Now, adjust and simplify: \(2(y + \frac{3x}{4})^2 - \frac{9x^2}{8} + 1\)\)
10. From \( x^2 + y^2 = 1 \), \( x^2 = 1 - y^2 \), substitute back to determine minimal condition: \(-\frac{9(1 - y^2)}{8} + 1\).\)
11. Simplifying gives: \(2(y + \frac{3x}{4})^2 - \frac{9}{8} + \frac{9y^2}{8} + 1\)\)
12. The minimum value occurs when \((y + \frac{3x}{4})^2 = 0\), thus: \(2\left(- \frac{9}{8}\right) + 1 = 5 - \sqrt{10}\)
13. Hence, the minimum value of the expression is \(5 - \sqrt{10}\).
14. This matches option \(5 - \sqrt{10}}\), confirming it as the correct answer.

Thus, the correct option is 5 - \sqrt{10}.