Question

Find the reference angle of each angle. $$ \frac{19 \pi}{6} $$

Short answer

\( \frac{\pi}{6} \)

Step-by-step solution

Convert radians to degrees if needed

Convert the angle \( \frac{19 \pi}{6} \) from radians to degrees to make it easier to find the reference angle. Use the conversion factor \( 180^\circ = \pi \text{ radians} \). Multiply: \[\frac{19 \pi}{6} \times \frac{180^\circ}{\pi} = \frac{19 \times 180^\circ}{6} = 570^\circ \]

Subtract multiples of 360 degrees

Since 570° is greater than 360°, subtract 360° to find the equivalent angle within one circle: \[570^\circ - 360^\circ = 210^\circ.\]

Determine the reference angle

The angle 210° is in the third quadrant. The reference angle for an angle in the third quadrant is \ ( \text{angle} - 180^\circ ) \. Compute: \[210^\circ - 180^\circ = 30^\circ.\]

Convert the reference angle back to radians

Now, convert the reference angle back to radians if needed: \[30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6}.\]

Key concepts

Radians to Degrees Conversion

When working with angles, it's often helpful to convert between radians and degrees. This can make calculations simpler. The relationship between radians and degrees is given by: \( 180^\text{°} = \pi \text{ radians} \). To convert radians to degrees, use the formula: \ \ \text{angle in degrees} = \text{angle in radians} \times \( \frac{180^\text{°}}{\text{\pi}} \).
For example, let's convert \ \( \frac{19\pi}{6} \ \) radians to degrees. Multiply by \( \frac{180^\text{°}}{\text{\pi}} \). The \ \pi \ \ symbols cancel out, so you calculate: \[ \frac{19 \pi}{6} \times \frac{180^\text{°}}{\text{\pi}} = \frac{19 \times 180^\text{°}}{6} = 570^\text{°} \]. This gives you a more familiar degree measure to work with.

Angle Subtraction

Sometimes converting to degrees results in an angle greater than 360°. In this case, you should subtract multiples of 360° until you get an angle between 0° and 360°.
For example, with the angle 570°: To reduce it, subtract 360°: \(570^\text{°} - 360^\text{°} = 210^\text{°} \). This step doesn't change the angle’s measure, it just simplifies it to one rotation within the circle, making it easier to find reference angles and work within the angle’s quadrant.

Quadrants

The coordinate plane is divided into four quadrants, and each quadrant has specific characteristics for angles.
Here's a brief overview:
- Quadrant I: 0° to 90°
- Quadrant II: 90° to 180°
- Quadrant III: 180° to 270°
- Quadrant IV: 270° to 360°
Knowing which quadrant an angle belongs to helps you find its reference angle. For instance, 210° falls in Quadrant III. The reference angle is the difference between the angle and 180°.

Radians to Degrees Back Conversion

Often, you'll need to convert your final reference angle back to radians. This is especially important in trigonometry and calculus.
The conversion is straightforward: Multiply the angle in degrees by \( \frac{\pi}{180^\text{°}} \).
Let's convert 30° to radians: \(30^\text{°} \times \frac{\pi}{180^\text{°}} = \frac{\pi}{6} \).
This step is crucial for answers in their original format.