To find the exact value of [tex]\(\sin(-330^\circ)\)[/tex], we'll follow these steps:
### Step 1: Find a Coterminal Angle
A coterminal angle is an angle that ends up at the same position as the original angle but is within the standard range from [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]. To find a positive coterminal angle for [tex]\(-330^\circ\)[/tex], add [tex]\(360^\circ\)[/tex]:
[tex]\[
-330^\circ + 360^\circ = 30^\circ
\][/tex]
So, the angle [tex]\(-330^\circ\)[/tex] is coterminal with [tex]\(30^\circ\)[/tex].
### Step 2: Verify if the Coterminal Angle is a Reference Angle
Since the coterminal angle [tex]\(30^\circ\)[/tex] is within the range from [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex], it is already in its simplest form. Thus, the reference angle is also [tex]\(30^\circ\)[/tex].
### Step 3: Find the Sine of the Reference Angle
Now, calculate the sine of the reference angle:
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
Therefore, the sine of the angle [tex]\(30^\circ\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
### Conclusion
The reference choice should be:
A. The angle [tex]\(-330^\circ\)[/tex] is a quadrantal angle. The angle [tex]\(-330^\circ\)[/tex] is coterminal with the angle [tex]\(30^\circ\)[/tex].
Thus, the exact value of [tex]\(\sin(-330^\circ)\)[/tex] is:
[tex]\[
\sin(-330^\circ) = \frac{1}{2}
\][/tex]
