Here's a "cleaned up" copy of the complete unit circle whose points you filled out today:

A few important things to remember:

- The only five magnitudes that ever show up as coordinates for these "nice" points are (in order from least to greatest): \[0,\ \dfrac12,\ \dfrac{\sqrt2}2,\ \dfrac{\sqrt3}2,\ 1\] Use negative signs where needed.
- The numbers \(0\) and \(1\) are always paired up, as are \(\dfrac12\) and \(\dfrac{\sqrt3}2\), while \(\dfrac{\sqrt2}2\) is always paired with itself.
- Instead of \(\dfrac{\sqrt{2}}{2}\), you can just as easily write \(\dfrac1{\sqrt2}\) instead — either one is as good as the other. There's no hard-and-fast "rule" that says you can
**never**have a radical in the denominator! It all depends on what you want to do with it. Choose the form that lends itself best to the situation.- If you're curious, one reason that
**historically**we used to rationalize the denominator has to do with by-hand calculations. If you want to calculate \(\dfrac1{\sqrt2}\) by hand, that's about \(1\div 1.414\) — set up that division problem and before long you'll see why it's unwieldy. But if you use the equivalent form \(\dfrac{\sqrt2}2\), you can see it's much easier to calculate as about \(0.707\).

- If you're curious, one reason that
- Don't bother using silly mnemonics to memorize the quadrants in which
**sine**and**cosine**are positive or negative. Instead, remember what these things**mean**(**height**and**overness**respectively) and use your eyes (or your mind's eye) to look for relationships.**There's no need to***memorize*it if you can*visualize*it.

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