Online Math Tutor LLC

Discover the Path to Mathematical Mastery 
An Education That Counts

Online Math Tutor LLC

Discover the Path to Mathematical Mastery 
An Education That Counts

Angles

Angle \(\theta\) satisfies

  \(0<\theta<90\) degrees

Angle \(\theta\) satisfies

  \(\theta=90\) degrees

Angle \(\theta\) satisfies

  \(90<\theta<180\) degrees

A complete rotation around a circle is \(2\pi\) radians.

To convert from Radians to Degrees use the multiplicative conversation factor.

\(\displaystyle\frac{180 \text{ degrees}}{\pi \text{ radians}}\)

A complete rotation around a circle is \(180\) degrees.

To convert from Degrees to Radians use the multiplicative  conversation factor.

\(\displaystyle\frac{\pi\text{ radians}}{180\text{ degrees}}\)

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.