Evaluate \(\frac{1}{1 - \sin 60°}\), leaving your answer in surd form.

Evaluate \(\frac{1}{1 - \sin 60°}\), leaving your answer in surd form.

Answer Details

We know that \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). Substituting this value in the given expression, we get: \[\frac{1}{1 - \sin 60^{\circ}} = \frac{1}{1 - \frac{\sqrt{3}}{2}}\] Rationalizing the denominator by multiplying both numerator and denominator by \(1 + \frac{\sqrt{3}}{2}\), we get: \begin{align*} \frac{1}{1 - \sin 60^{\circ}} &= \frac{1}{1 - \frac{\sqrt{3}}{2}} \cdot \frac{1 + \frac{\sqrt{3}}{2}}{1 + \frac{\sqrt{3}}{2}} \\ &= \frac{1 + \frac{\sqrt{3}}{2}}{1 - \frac{3}{4}} \\ &= \frac{1 + \frac{\sqrt{3}}{2}}{\frac{1}{4}} \\ &= 4 + 2\sqrt{3} \end{align*} Therefore, the answer is \boxed{4 + 2\sqrt{3}}.