Evaluate \(\frac{2\sin 30 + 5\tan 60}{\sin 60}\), leaving your answer in surd form.
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Correct Answer: Option A
Explanation:
\(\frac{2\sin 30 + 5\tan 60}{\sin 60}\)
\(\sin 30 = \frac{1}{2}; \tan 60 = \sqrt{3}; \sin 60 = \frac{\sqrt{3}}{2}\)
\(\therefore \frac{2\sin 30 + 5\tan 60}{\sin 60} = \frac{2(\frac{1}{2}) + 5(\sqrt{3})}{\frac{\sqrt{3}}{2}}\)\)
= \(\frac{1 + 5\sqrt{3}}{\frac{\sqrt{3}}{2}}\)
= \(\frac{2(1 + 5\sqrt{3})}{\sqrt{3}}\)
= \(\frac{2 + 10\sqrt{3}}{\sqrt{3}}\)
Rationalizing, we get
= \(\frac{2\sqrt{3} + 30}{3}\)
= \(\frac{2}{3} \sqrt{3} + 10\)
\(\frac{2\sin 30 + 5\tan 60}{\sin 60}\)
\(\sin 30 = \frac{1}{2}; \tan 60 = \sqrt{3}; \sin 60 = \frac{\sqrt{3}}{2}\)
\(\therefore \frac{2\sin 30 + 5\tan 60}{\sin 60} = \frac{2(\frac{1}{2}) + 5(\sqrt{3})}{\frac{\sqrt{3}}{2}}\)\)
= \(\frac{1 + 5\sqrt{3}}{\frac{\sqrt{3}}{2}}\)
= \(\frac{2(1 + 5\sqrt{3})}{\sqrt{3}}\)
= \(\frac{2 + 10\sqrt{3}}{\sqrt{3}}\)
Rationalizing, we get
= \(\frac{2\sqrt{3} + 30}{3}\)
= \(\frac{2}{3} \sqrt{3} + 10\)