Section 5.1 The Unit Circle
The first key to understanding trigonometry is to know the unit circle. The unit circle is the circle centered at (0,0) with radius 1.
Consider an angle \(\theta\) in the unit circle. The angle is positive if it is measured counterclockwise from the positive x-axis and negative if it is measured clockwise.
The above angles are measured using degrees. An angle \(\theta\) may also be measured using radians. The radian measurement corresponds to a distance around the circumference, \(C\text{,}\) of the unit circle (\(C' = 2\pi\)).
Let us measure an arc on the unit circle starting at (1,0) of length \(\frac{\pi}{4}\) and ending at a point \(P\text{.}\) If we draw a ray from the origin through point \(P\text{,}\) we have formed an angle \(\theta\text{,}\) where \(\theta\) = \(\frac{\pi}{4}\)radians.
The following two angles are the same
To convert radians to degrees and vice versa, use the following equation.
\begin{equation*}
\pi \text{ radians} = 180^\circ
\end{equation*}
Convert \(30^\circ\) to radians.
\begin{equation*}
\pi \text{ radians} = 180^\circ
\end{equation*}
\begin{equation*}
\frac{\pi}{180} \text{ radians} = 1^\circ
\end{equation*}
\begin{equation*}
30\cdot\frac{\pi}{180} \text{ radians} = 30\cdot 1^\circ
\end{equation*}
\begin{equation*}
\frac{\pi}{6} \text{ radians} = 30^\circ
\end{equation*}
Example 5.1.6. convert radians to degrees.
Convert \(\frac{8\pi}{5}\) radians to degrees.
Solution.
\begin{equation*}
\pi \text{ radians} = 180^\circ
\end{equation*}
\begin{equation*}
\frac{8}{5}\cdot\pi \text{ radians } = \frac{8}{5}\cdot 180^\circ
\end{equation*}
\begin{equation*}
\frac{8\pi}{5} \text{ radians } = 288^\circ
\end{equation*}
