To solve this problem, let's break it down step-by-step.
1. Measure of angle [tex]\(X\)[/tex] in radians:
The measure of angle [tex]\(X\)[/tex] is given as [tex]\(\frac{\pi}{6}\)[/tex] radians.
2. Measure of angle [tex]\(Y\)[/tex] in radians:
Angle [tex]\(Y\)[/tex] is [tex]\(\frac{2\pi}{3}\)[/tex] radians greater than angle [tex]\(X\)[/tex]. So, we need to add [tex]\(\frac{2\pi}{3}\)[/tex] to [tex]\(\frac{\pi}{6}\)[/tex]:
[tex]\[
\text{angle } Y = \frac{\pi}{6} + \frac{2\pi}{3}
\][/tex]
3. Finding a common denominator:
To add these fractions, we need a common denominator. The least common multiple of 6 and 3 is 6.
[tex]\[
\frac{2\pi}{3} = \frac{2\pi \cdot 2}{3 \cdot 2} = \frac{4\pi}{6}
\][/tex]
4. Adding the angles:
Now we add the fractions:
[tex]\[
\frac{\pi}{6} + \frac{4\pi}{6} = \frac{1\pi + 4\pi}{6} = \frac{5\pi}{6}
\][/tex]
Therefore, the measure of angle [tex]\(Y\)[/tex] is [tex]\(\frac{5\pi}{6}\)[/tex] radians.
5. Converting radians to degrees:
To convert from radians to degrees, we use the fact that [tex]\( \pi \)[/tex] radians is equal to 180 degrees. The conversion factor is:
[tex]\[
1 \text{ radian} = \frac{180}{\pi} \text{ degrees}
\][/tex]
Thus,
[tex]\[
\text{angle } Y \text{ in degrees} = \frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \cdot 180}{6} = \frac{900}{6} = 150 \text{ degrees}
\][/tex]
So, the measure of angle [tex]\(Y\)[/tex] is 150 degrees.
Answer: D) 150
