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Section 2.1 Angles

Watermark text: DRAFT

Subsection 2.1.1 Coterminal Angles

Definition 2.1.1.

Two angles are coterminal if they have the same terminal side when in standard position.

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.

This is a graph showing coterminal angles of 45 degrees, 405 degrees, and -315 degrees which all have the same terminal side.
Figure 2.1.3. Coterminal angles

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.

Graph showing the angle pi/3 radians.
Figure 2.1.6.
Graph showing the angle 30 degrees.
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).

Graph showing the angle (2pi)/3 radians which has a reference angle of pi/3 radians.
Figure 2.1.9.
Graph showing the angle 150 degrees which has a reference angle of 30 degrees.
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).

Graph showing the angle (4pi)/3 radians which has a reference angle of pi/3 radians.
Figure 2.1.12.
Graph showing the angle 210 degrees which has a reference angle of 30 degrees.
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).

Graph showing the angle (5pi)/3 radians which has a reference angle of pi/3 radians.
Figure 2.1.15.
Graph showing the angle 330 degrees which has a reference angle of 30 degrees.
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).

Graph showing the angle 7.5 radians which has a reference angle of 1.2 radians.
Figure 2.1.18.
Graph showing the angle -137 degrees which has a reference angle of 43 degrees.
Figure 2.1.19.

Exercises 2.1.3 Exercises

Find both a positive and negative angle that is coterminal angle with the following angles.

1.

\(63^{\circ}\)

2.

\(\frac{\pi}{9}\)

3.

\(\frac{13\pi}{8}\)

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.

\(-100^{\circ}\)