To find the exact value of [tex]\(\sin(-330^\circ)\)[/tex], let's proceed step-by-step through the problem.
1. Determine the Coterminal Angle:
A coterminal angle of [tex]\(-330^\circ\)[/tex] is an angle that lies between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex]. To find this, we add [tex]\(360^\circ\)[/tex] to [tex]\(-330^\circ\)[/tex] (since adding a full rotation does not change the sine value):
[tex]\[
-330^\circ + 360^\circ = 30^\circ
\][/tex]
Thus, [tex]\(-330^\circ\)[/tex] is coterminal with [tex]\(30^\circ\)[/tex].
2. Check if [tex]\(-330^\circ\)[/tex] is a Quadrantal Angle:
A quadrantal angle is one that lies on the x-axis or y-axis (multiples of [tex]\(90^\circ\)[/tex]). Since [tex]\(-330^\circ\)[/tex] or [tex]\(30^\circ\)[/tex] does not lie on the axes, it is not a quadrantal angle.
Therefore, Choice B is correct:
[tex]\[
\text{The angle } -330^\circ \text{ is not a quadrantal angle. The reference angle for } -330^\circ \text{ is } 30^\circ.
\][/tex]
3. Find the Exact Value of [tex]\(\sin(-330^\circ)\)[/tex]:
Since [tex]\(-330^\circ\)[/tex] is coterminal with [tex]\(30^\circ\)[/tex], we can find [tex]\(\sin(-330^\circ)\)[/tex] by finding [tex]\(\sin(30^\circ)\)[/tex].
The sine of [tex]\(30^\circ\)[/tex] is a well-known trigonometric value:
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
4. Consider the Sign of the Sine Function:
The angle [tex]\(-330^\circ\)[/tex] lies in the fourth quadrant (as does [tex]\(30^\circ\)[/tex] because we rotated [tex]\(360^\circ\)[/tex] clockwise). In the fourth quadrant, the sine function is negative.
Therefore:
[tex]\[
\sin(-330^\circ) = -\sin(30^\circ) = -\frac{1}{2}
\][/tex]
So, the exact value of [tex]\(\sin(-330^\circ)\)[/tex] is:
[tex]\[
\boxed{-\frac{1}{2}}
\][/tex]
Hence, for the final part:
Choice A is correct:
[tex]\[
\sin(-330^\circ) = -\frac{1}{2}
\][/tex]
