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Unit Circle

Finding Patterns In The Unit Circle

Unit Circle

Unit Circle Related Diagrams

Location	\(\theta\)	\(\texttt{sin}(\theta)\)	\(\texttt{cos}(\theta)\)	\(\texttt{tan}(\theta)\)
x-axis	\(0\)	\(0\)	\(1\)	\(0\)
Quadrant 1	\(\displaystyle\frac{\pi}{6}\)	\(\displaystyle\frac{1}{2}\)	\(\displaystyle\frac{\sqrt{3}}{2}\)	\(\displaystyle\frac{\sqrt{3}}{3}\)
	\(\displaystyle\frac{\pi}{4}\)	\(\displaystyle\frac{\sqrt{2}}{2}\)	\(\displaystyle\frac{\sqrt{2}}{2}\)	\(\displaystyle 1\)
	\(\displaystyle\frac{\pi}{3}\)	\(\displaystyle\frac{\sqrt{3}}{2}\)	\(\displaystyle\frac{1}{2}\)	\(\displaystyle\sqrt{3}\)
y-axis	\(\displaystyle\frac{\pi}{2}\)	\(1\)	\(0\)	\(\texttt{DNE}\)

Location	\(\theta\)	\(\texttt{sin}(\theta)\)	\(\texttt{cos}(\theta)\)	\(\texttt{tan}(\theta)\)
y-axis	\(\displaystyle\frac{\pi}{2}\)	\(1\)	\(0\)	\(\texttt{DNE}\)
Quadrant 2	\(\displaystyle\frac{\pi}{3}\)	\(\displaystyle\frac{\sqrt{3}}{2}\)	\(\displaystyle-\frac{1}{2}\)	\(\displaystyle-\sqrt{3}\)
	\(\displaystyle\frac{\pi}{4}\)	\(\displaystyle\frac{\sqrt{2}}{2}\)	\(\displaystyle-\frac{\sqrt{2}}{2}\)	\(\displaystyle-1\)
	\(\displaystyle\frac{\pi}{6}\)	\(\displaystyle\frac{1}{2}\)	\(\displaystyle-\frac{\sqrt{3}}{2}\)	\(\displaystyle-\frac{\sqrt{3}}{3}\)
x-axis	\(\pi\)	\(0\)	\(1\)	\(0\)

Location	\(\theta\)	\(\texttt{sin}(\theta)\)	\(\texttt{cos}(\theta)\)	\(\texttt{tan}(\theta)\)
x-axis	\(\pi\)	\(0\)	\(-1\)	\(0\)
Quadrant 3	\(\displaystyle\frac{7\pi}{6}\)	\(\displaystyle\frac{1}{2}\)	\(\displaystyle-\frac{\sqrt{3}}{2}\)	\(\displaystyle\frac{\sqrt{3}}{3}\)
	\(\displaystyle\frac{5\pi}{4}\)	\(\displaystyle-\frac{\sqrt{2}}{2}\)	\(\displaystyle-\frac{\sqrt{2}}{2}\)	\(\displaystyle 1\)
	\(\displaystyle\frac{4\pi}{3}\)	\(\displaystyle-\frac{\sqrt{3}}{2}\)	\(\displaystyle-\frac{1}{2}\)	\(\displaystyle\sqrt{3}\)
y-axis	\(\displaystyle\frac{3\pi}{2}\)	\(-1\)	\(0\)	\(\texttt{DNE}\)

Location	\(\theta\)	\(\texttt{sin}(\theta)\)	\(\texttt{cos}(\theta)\)	\(\texttt{tan}(\theta)\)
y-axis	\(\displaystyle\frac{3\pi}{2}\)	\(-1\)	\(0\)	\(\texttt{DNE}\)
Quadrant 4	\(\displaystyle\frac{5\pi}{3}\)	\(\displaystyle-\frac{\sqrt{3}}{2}\)	\(\displaystyle\frac{1}{2}\)	\(\displaystyle-\sqrt{3}\)
	\(\displaystyle\frac{7\pi}{4}\)	\(\displaystyle-\frac{\sqrt{2}}{2}\)	\(\displaystyle\frac{\sqrt{2}}{2}\)	\(\displaystyle -1\)
	\(\displaystyle\frac{11\pi}{6}\)	\(\displaystyle-\frac{1}{2}\)	\(\displaystyle\frac{\sqrt{3}}{2}\)	\(\displaystyle-\frac{\sqrt{3}}{3}\)
x-axis	\(\displaystyle2\pi\)	\(0\)	\(1\)	\(0\)

*Be sure to take some time to memorize these special angle values on the unit circle.

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