# MTH 111–112 Supplement

## Section2.1Angles

### Subsection2.1.1Coterminal Angles

#### Definition2.1.1.

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.

#### Example2.1.2.

The angles $$45^{\circ}\text{,}$$ $$405^{\circ}\text{,}$$ and $$-315^{\circ}$$ are coterminal as illustrated in Figure 2.1.3.

### Subsection2.1.2Reference Angles

#### Definition2.1.4.

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.

#### Example2.1.5.

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.

#### Example2.1.8.

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).

#### Example2.1.11.

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).

#### Example2.1.14.

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).

#### Example2.1.17.

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).

### Subsection2.1.3Degrees, Minutes, and Seconds

When measuring angles in degrees, fractions of a degree can be represented in minutes and seconds.

#### Definition2.1.20.

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.

#### Example2.1.21.

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*}

#### Example2.1.22.

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*}

### Exercises2.1.4Exercises

#### 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.
$$0.416^{\circ}$$