## A mnemonic for remembering values on the unit circle

In an earlier post, I told you that you must memorize the values on the unit circle if you want to be great at trigonometry. So let’s say that you have taken my advice, but on a test you have a big brain cramp and you forget the values. Here’s a trick that will help you fill in the values in the first quadrant. The five angles in the first quadrant are 0°, 30°, 45°, 60°, and 90°. And the sines of these angles just happen to be:

$\underline{ \qquad \theta \qquad \sin \theta \qquad \qquad \qquad}$

$0 \textdegree \qquad \; \; 0 \qquad =\dfrac{\sqrt{0}}{2}$

$30 \textdegree \qquad \dfrac{1}{2} \qquad = \dfrac{\sqrt{1}}{2}$

$45 \textdegree \qquad \dfrac{\sqrt{2}}{2} \quad = \dfrac{\sqrt{2}}{2}$

$60 \textdegree \qquad \dfrac{\sqrt{3}}{2} \quad = \dfrac{\sqrt{3}}{2}$

$90 \textdegree \qquad 1 \qquad = \dfrac{\sqrt{4}}{2}$

Do you see the pattern in the last column? The sines increase from

$\dfrac{\sqrt{0}}{2}$ to $\dfrac{\sqrt{4}}{2}$

That’s an easy pattern to remember even during the worst brain cramp. Why does it work? It’s just a coincidence. But if you use this pattern, you can fill in the sine values in the first quadrant. Then the cosine values are the same in reverse order. Once you’ve completed the first quadrant, you can fill in the rest of the unit circle by using reference angles. Now you have a completed unit circle!

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