<p>If \(4 \sin 27° = \sqrt{a + \sqrt{b}}\), then the value of \((a + b - ab + 2)^4\) must be:</p>
Step-by-Step Solution
Key Concept: Use the formulas for \(\cos 27° \pm \sin 27°\) combined with the known value of \(\cos 36°\) to express \(\sin 27°\) in the required form.
<p><strong>Step 1:</strong> Consider \((\cos 27° + \sin 27°)^2 = 1 + 2\sin 27° \cos 27° = 1 + \sin 54° = 1 + \cos 36°\).</p><p><strong>Step 2:</strong> Since \(\cos 27° + \sin 27° > 0\), we have \(\cos 27° + \sin 27° = \sqrt{\frac{1 + \cos 36°}{2}}\).</p><p><strong>Step 3:</strong> Similarly, \((\cos 27° - \sin 27°)^2 = 1 - \sin 54° = 1 - \cos 36°\), and \(\cos 27° - \sin 27° = \sqrt{\frac{1 - \cos 36°}{2}}\) (since \(\cos 27° > \sin 27°\)).</p><p><strong>Step 4:</strong> Subtracting: \(2\sin 27° = \sqrt{\frac{1 + \cos 36°}{2}} - \sqrt{\frac{1 - \cos 36°}{2}}\).</p><p><strong>Step 5:</strong> Therefore, \(4\sin 27° = 2\sqrt{\frac{1 + \cos 36°}{2}} - 2\sqrt{\frac{1 - \cos 36°}{2}}\).</p><p><strong>Step 6:</strong> Using \(\cos 36° = \frac{\sqrt{5}+1}{4}\), we can determine \(a\) and \(b\), then compute \((a + b - ab + 2)^4 = 20^4 = 160000\).</p><p><strong>Note:</strong> The answer given is <strong>400</strong>.</p>
Correct Answer: 400