Definition 2.1.1.
Two angles are coterminal if they have the same terminal side when in standard position.
Section 2.1 Angles
Subsection 2.1.1 Coterminal Angles
Since \(360^{\circ}\) represents a complete revolution, if we add integer-multiples of \(360^{\circ}\) to an angle measured in degrees, we’ll obtain a coterminal angle. Similarly, since \(2\pi\) represents a complete revolution in radians, if we add integer-multiples of \(2\pi\) to an angle measured in radians, we’ll obtain a coterminal angle. We can summarize this information as follows
If \(\theta\) is measured in degrees, \(\theta\) and \(\theta + 360^{\circ} \cdot k\text{,}\) where \(k\in\mathbb{Z}\text{,}\) are coterminal.
If \(\theta\) is measured in radians, \(\theta\) and \(\theta + 2\pi \cdot k\text{,}\) where \(k\in\mathbb{Z}\text{,}\) are coterminal.
Example 2.1.2.
The angles \(45^{\circ}\text{,}\) \(405^{\circ}\text{,}\) and \(-315^{\circ}\) are coterminal as illustrated in Figure 2.1.3.
Subsection 2.1.2 Reference Angles
Definition 2.1.4.
The reference angle for an angle in standard position is the positive acute angle formed by the \(x\)-axis and the terminal side of the angle.
Depending on the location of the angle’s terminal side, we’ll have to use a different calculation to determine the angle’s reference angle.
Example 2.1.5.
The angles \(\frac{\pi}{3}\) and \(30^{\circ}\) are their own reference angles since they are acute angles; seen in Figure 2.1.6 and Figure 2.1.7.
Example 2.1.8.
The reference angle for \(\frac{2\pi}{3}\) is \(\pi-\frac{2\pi}{3}=\frac{\pi}{3}\) (see Figure 2.1.9), while the reference angle for \(150^{\circ}\) is \(180^{\circ}-150^{\circ}=30^{\circ}\) (see Figure 2.1.10).
Example 2.1.11.
The reference angle for \(\frac{4\pi}{3}\) is \(\frac{4\pi}{3}-\pi=\frac{\pi}{3}\) (see Figure 2.1.12), while the reference angle for \(210^{\circ}\) is \(210^{\circ}-180^{\circ}=30^{\circ}\) (see Figure 2.1.13).
Example 2.1.14.
The reference angle for \(\frac{5\pi}{3}\) is \(2\pi-\frac{5\pi}{3}=\frac{\pi}{3}\) (see Figure 2.1.15), while the reference angle for \(330^{\circ}\) is \(360^{\circ}-330^{\circ}=30^{\circ}\) (see Figure 2.1.16).
Example 2.1.17.
The reference angle for \(7.5\) radians is \(7.5-2\pi\approx 1.2\) radians (see Figure 2.1.18), and the reference angle for \(-137^{\circ}\) is \(180^{\circ}+( \, -137^{\circ} ) \,=43^{\circ}\) (see Figure 2.1.19).
Subsection 2.1.3 Degrees, Minutes, and Seconds
When measuring angles in degrees, fractions of a degree can be represented in minutes and seconds.
Definition 2.1.20.
One minute is \(\frac{1}{60}\) of a degree, so \(60\) minutes is equal to one degree (written \(60'=1^{\circ}\)).
One second is \(\frac{1}{60}\) of a minute, i.e., \(\frac{1}{3600}\) of a degree, so \(60\) seconds is equal to one minute (written \(60''=1'\)) and \(3600\) seconds is equal to one degree (written \(3600''=1^{\circ}\)).
When we write an angle’s measure in the form \(D^{\circ}M'S''\) where (\(D\text{,}\) \(M\text{,}\) and \(S\) are real numbers), then the angle’s measure is \(D\) degrees plus \(M\) minutes plus \(S\) seconds.
Example 2.1.21.
Convert \(34^{\circ}15'27''\) into decimal form.
Solution.
\begin{align*}
34^{\circ}15'27'' \amp= 34^{\circ}+15'\left( \frac{1^{\circ}}{60'} \right)+27'' \left(\frac{1^{\circ}}{3600''}\right) \\
\amp= 34^{\circ}+0.25^{\circ}+0.0075^{\circ} \\
\amp= 34.2575^{\circ}
\end{align*}
Example 2.1.22.
Convert \(61.72^{\circ}\) into \(D^{\circ}M'S''\) form.
Solution.
\begin{align*}
61.72^{\circ} \amp= 61^{\circ}+0.72^{\circ} \\
\amp= 61^{\circ}+0.72^{\circ} \cdot \left( \frac{60'}{1^{\circ}} \right) \\
\amp= 61^{\circ}+43.2' \\
\amp= 61^{\circ}43'+0.2' \\
\amp= 61^{\circ}43'+0.2' \cdot \left( \frac{60''}{1'} \right) \\
\amp= 61^{\circ}43'12''
\end{align*}
Exercises 2.1.4 Exercises
Coterminal Angles.
In Exercises 1–3, find both a positive and negative angle that is coterminal angle with the following angles.
1.
\(63^{\circ}\)
2.
\(\frac{\pi}{9}\)
3.
\(\frac{13\pi}{8}\)
Reference Angles.
In Exercises 4–12, find the reference angle for the following angles.
4.
\(120^{\circ}\)
5.
\(\frac{5\pi}{4}\)
6.
\(400^{\circ}\)
7.
\(\frac{13\pi}{8}\)
8.
\(2\)
9.
\(\frac{10\pi}{11}\)
10.
\(2000^{\circ}\)
11.
\(-\frac{9\pi}{5}\)
12.
\(-100^{\circ}\)
Convert to Decimal Form.
In Exercises 13–16, convert the angle measure to decimal form (round your answers to the nearest thousandth when necessary).
13.
\(243^{\circ}10'\)
14.
\(3^{\circ}25'\)
15.
\(-23^{\circ}3'\)
16.
\(75^{\circ}32'17''\)
Convert to \(D^{\circ}M'S''\) Form.
In Exercises 17–20, convert the angle measure to \(D^{\circ}M'S''\) form.
17.
\(12.4^{\circ}\)
18.
\(1.53^{\circ}\)
19.
\(-144.9^{\circ}\)
20.
\(0.416^{\circ}\)
