Find the exact value of the expression. Do not use a calculator.

[tex]\[ \sin \left(-330^{\circ}\right) \][/tex]

Determine the reference angle or coterminal angle for [tex]\(-330^{\circ}\)[/tex]. Select the correct choice below and fill in the answer box to complete your choice.

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]. [tex]\(\square\)[/tex]
(Use angle measures greater than or equal to 0 and less than 360.)

B. The angle [tex]\(-330^{\circ}\)[/tex] is not a quadrantal angle. The reference angle for [tex]\(-330^{\circ}\)[/tex] is [tex]\(\square\)[/tex].

(Note: The correct reference angle or coterminal angle should be written in the answer box.)



Answer :

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]