Discussion
If \(4\sin^{2}x - 2(1 + \sqrt{3})\sin x + \sqrt{3} = 0\) and \(15^{\circ} \lt x \lt 150^{\circ}\) , then the values of \(x\) are
Answer: C
The values are 30°, 60°, 120°.
Treat the equation as a quadratic in \(s = \sin x\). It factorises:
\[ \begin{aligned} 4s^{2} - 2(1 + \sqrt{3})s + \sqrt{3} &= 0 \\ (2s - 1)(2s - \sqrt{3}) &= 0 \end{aligned} \]
So \(\sin x = \frac{1}{2}\) or \(\sin x = \frac{\sqrt{3}}{2}\). In \(15^{\circ} \lt x \lt 150^{\circ}\): \(\sin x = \frac{1}{2}\) gives \(x = 30^{\circ}\) (150° is excluded by the strict inequality), and \(\sin x = \frac{\sqrt{3}}{2}\) gives \(x = 60^{\circ}\) and \(120^{\circ}\).
45° and 90° give \(\sin x = \frac{1}{\sqrt{2}}\) and 1, which are not roots, so every other option fails.
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