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Precalculus Review Materials

Jane Butterfield, Gary MacGillivray

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.
The unit circle centered at (0,0) with radius 1, drawn in the cartesian plane.
Figure 5.1.1.
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.
described in detail following the image
The angles \(135^\circ\) and \(-135^\circ\) drawn on the unit circle.
Figure 5.1.2.
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.
described in detail following the image
The angle \(\pi/4\) drawn on the unit circle.
Figure 5.1.3.
The following two angles are the same
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The angle \(45^\circ\) drawn on one copy of the unit circle, and \(\pi/4\) drawn on another copy.
Figure 5.1.4.
To convert radians to degrees and vice versa, use the following equation.
\begin{equation*} \pi \text{ radians} = 180^\circ \end{equation*}

Example 5.1.5. convert degrees to radians.

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*}
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