<s>[INST] How does $\frac{\sqrt{3}}{2} = \sqrt{\frac{3}{4}}$? I'm studying trigonometry and one of the key triangles (60 30 90) has a side of $\frac{\sqrt{3}}{2}$. I understand that it is derived from an equilateral triangle of sides 1, however I can't see how $\frac{\sqrt{3}}{2} = \sqrt{\frac{3}{4}}$ and why  $\frac{\sqrt{3}}{2}$ is simpler to use than $\sqrt{\frac{3}{4}}$. I obtained $\sqrt{\frac{3}{4}}$ from $\sqrt{1^2 - (\frac{1}{2})^2}$
 [/INST] $$\sqrt{\frac{3}{4}}=\sqrt{3\cdot4^{-1}}=\sqrt{3\cdot2^{-2}}=\sqrt{3}\cdot2^{-1}=\frac{\sqrt{3}}{2}$$
As to which to use, normally people tend to like to have as little surds as possible.
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