The Circular Functions (Wrap-Up)

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

Click the picture to enlarge it, especially if it's blurry.

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): $$0,{\displaystyle \frac{1}{2}},{\displaystyle \frac{\sqrt{2}}{2}},{\displaystyle \frac{\sqrt{3}}{2}},1$$ Use negative signs where needed.
• The numbers $0$ and $1$ are always paired up, as are ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{\sqrt{3}}{2}}$, while ${\displaystyle \frac{\sqrt{2}}{2}}$ is always paired with itself.
• Instead of ${\displaystyle \frac{\sqrt{2}}{2}}$, you can just as easily write ${\displaystyle \frac{1}{\sqrt{2}}}$ 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 ${\displaystyle \frac{1}{\sqrt{2}}}$ by hand, that's about $1÷1.414$ — set up that division problem and before long you'll see why it's unwieldy. But if you use the equivalent form ${\displaystyle \frac{\sqrt{2}}{2}}$, you can see it's much easier to calculate as about $0.707$.
• 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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Posted by Bill Shillito