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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]\(\square\)[/tex] [tex]\(^{\circ}\)[/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] [tex]\(^{\circ}\)[/tex].

Find the exact value of [tex]\(\sin \left(-330^{\circ}\right)\)[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]\(\sin \left(-330^{\circ}\right) =\)[/tex] [tex]\(\square\)[/tex] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

B. The value is undefined.


Sagot :

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]