To solve the problem, let's break it down into several steps.
### Step 1: Find the Coterminal Angle
To find an angle coterminal with [tex]\(-330^\circ\)[/tex] that lies between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex]:
[tex]\[ \text{Coterminal angle} = -330^\circ + 360^\circ = 30^\circ \][/tex]
Therefore, [tex]\(-330^\circ\)[/tex] is coterminal with [tex]\(30^\circ\)[/tex].
### Step 2: Determine if the Angle is Quadrantal
A quadrantal angle is an angle whose terminal side lies on an axis (i.e., [tex]\(0^\circ\)[/tex], [tex]\(90^\circ\)[/tex], [tex]\(180^\circ\)[/tex], or [tex]\(270^\circ\)[/tex]). Since [tex]\(30^\circ\)[/tex] is not one of these angles, [tex]\(-330^\circ\)[/tex] is not a quadrantal angle.
### Step 3: Find the Reference Angle
For an angle given in standard position, the reference angle is the acute angle (the smallest positive angle) formed by the terminal side of the given angle and the x-axis. Since the coterminal angle [tex]\(30^\circ\)[/tex] is already an acute angle in the first quadrant, it is its own reference angle.
[tex]\[ \text{Reference angle} = 30^\circ \][/tex]
Thus, the correct choice is:
[tex]\[ \text{B. The angle } -330^\circ \text{ is not a quadrantal angle. The reference angle for } -330^\circ \text{ is } 30^\circ. \][/tex]
### Step 4: Find the Exact Value of [tex]\(\sin(-330^\circ)\)[/tex]
Since we now know that [tex]\(-330^\circ\)[/tex] is coterminal with [tex]\(30^\circ\)[/tex], we can use the sine value of [tex]\(30^\circ\)[/tex]. The sine of an angle in the fourth quadrant (where [tex]\(-330^\circ\)[/tex] terminates) is negative, and since [tex]\(30^\circ\)[/tex] refers to the same position but in the positive direction:
[tex]\[ \sin(30^\circ) = \frac{1}{2} \][/tex]
Since [tex]\(-330^\circ\)[/tex] is in the fourth quadrant, we have:
[tex]\[ \sin(-330^\circ) = -\sin(30^\circ) = -\frac{1}{2} \][/tex]
Thus, the correct choice is:
[tex]\[ \text{A. } \sin(-330^\circ) = -\frac{1}{2} \][/tex]
