To determine the reference angle for an angle measuring [tex]\(150^\circ\)[/tex], you need to understand that the reference angle is the smallest angle that the given angle makes with the x-axis.
When dealing with angles greater than [tex]\(90^\circ\)[/tex] and less than [tex]\(180^\circ\)[/tex], the angle lies in the second quadrant. In the second quadrant, the reference angle is found by taking the difference between [tex]\(180^\circ\)[/tex] and the given angle since [tex]\(180^\circ\)[/tex] is the straight line that separates the first two quadrants.
Given the angle [tex]\(150^\circ\)[/tex]:
1. Identify that the angle is in the second quadrant since it is greater than [tex]\(90^\circ\)[/tex] but less than [tex]\(180^\circ\)[/tex].
2. The formula to find the reference angle for angles in the second quadrant is [tex]\(180^\circ - x\)[/tex], where [tex]\(x\)[/tex] is the given angle.
Substitute the given angle, [tex]\(150^\circ\)[/tex]:
[tex]\[ 180^\circ - 150^\circ \][/tex]
Calculate the difference:
[tex]\[ 180^\circ - 150^\circ = 30^\circ \][/tex]
Therefore, the reference angle for an angle measuring [tex]\(150^\circ\)[/tex] is [tex]\(30^\circ\)[/tex], and the correct expression to determine the reference angle is:
[tex]\[ 180^\circ - x \][/tex]
