Use the \(30-60-90\) side ratio \(1:\sqrt{3}:2\) to derive \(\sin\), \(\cos\), and \(\tan\) for \(30°\), followed by \(\sin\), \(\cos\), and \(\tan\) for \(60°\). Simplify every exact value.

The same thirty-sixty-ninety triangle yields two sets of trig values because opposite and adjacent swap when the reference angle changes. We’ll derive sine, cosine, and tangent for the first angle, rationalize as needed, reverse the leg roles for the complement, and audit all six labels.