To find the quadrant in which the terminal ray of the angle [tex]\(-\frac{10}{9}\pi\)[/tex] lies, we need to convert the angle to degrees and then determine its equivalent positive angle.
1. Convert the angle from radians to degrees:
- The angle in degrees can be calculated from radians by using the formula:
[tex]\[
\text{Angle in degrees} = \text{Angle in radians} \times \frac{180}{\pi}
\][/tex]
- Substituting [tex]\(-\frac{10}{9}\pi\)[/tex] into the formula:
[tex]\[
\text{Angle in degrees} = -\frac{10}{9} \times 180 = -200\text{ degrees}
\][/tex]
2. Determine the equivalent positive angle:
- To find the equivalent positive angle, add 360 degrees until the angle is within the range of 0 to 360 degrees:
[tex]\[
-200 + 360 = 160\text{ degrees}
\][/tex]
3. Determine the quadrant:
- Dividing the full circle (360 degrees) into four quadrants:
- Quadrant I: [tex]\(0 \leq \theta < 90\)[/tex] degrees
- Quadrant II: [tex]\(90 \leq \theta < 180\)[/tex] degrees
- Quadrant III: [tex]\(180 \leq \theta < 270\)[/tex] degrees
- Quadrant IV: [tex]\(270 \leq \theta < 360\)[/tex] degrees
- Since [tex]\(160\)[/tex] degrees falls within the range of [tex]\(90 \leq \theta < 180\)[/tex] degrees, the terminal ray of the angle lies in Quadrant II.
4. State the angle measure:
- The original angle measured in degrees before any modifications were made is:
[tex]\[
-200\text{ degrees}
\][/tex]
Thus, the terminal ray of an angle measuring [tex]\(-\frac{10}{9} \pi\)[/tex] lies in the Quadrant II. This angle measures -200 degrees.
