To solve the equation [tex]\(\sin(90^\circ - x) = -\frac{\sqrt{3}}{2}\)[/tex], follow these steps:
1. Understand the Unit Circle:
- In the unit circle, the sine of an angle represents the y-coordinate of the corresponding point on the circle.
- The sine of an angle can take the value [tex]\(-\frac{\sqrt{3}}{2}\)[/tex] at specific points.
2. Determine the Angles:
- [tex]\(\sin(\theta) = -\frac{\sqrt{3}}{2}\)[/tex] occurs at exactly two angles in one rotation of the unit circle (0° to 360°): [tex]\(240^\circ\)[/tex] and [tex]\(300^\circ\)[/tex].
3. Examine the Given Equation:
- We have [tex]\(\sin(90^\circ - x) = -\frac{\sqrt{3}}{2}\)[/tex].
4. Identify the Corresponding Values for [tex]\(90^\circ - x\)[/tex]:
- Since [tex]\(\sin(90^\circ - x)\)[/tex] must equal [tex]\(-\frac{\sqrt{3}}{2}\)[/tex], we need [tex]\(90^\circ - x\)[/tex] to match one of the angles where [tex]\(\sin(\theta) = -\frac{\sqrt{3}}{2}\)[/tex].
5. Match Angles:
- The angles where [tex]\(\sin(\theta) = -\frac{\sqrt{3}}{2}\)[/tex] are [tex]\(240^\circ\)[/tex] and [tex]\(300^\circ\)[/tex].
- Therefore, [tex]\(90^\circ - x = 240^\circ\)[/tex] or [tex]\(90^\circ - x = 300^\circ\)[/tex].
6. Solve for [tex]\(x\)[/tex]:
- For [tex]\(90^\circ - x = 240^\circ\)[/tex]:
[tex]\[
90^\circ - x = 240^\circ \implies -x = 240^\circ - 90^\circ \implies -x = 150^\circ \implies x = -150^\circ
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\sin(90^\circ - x) = -\frac{\sqrt{3}}{2}\)[/tex] is [tex]\(-150^\circ\)[/tex].
Therefore, the correct option from the drop-down menu for the value of [tex]\(x\)[/tex] is [tex]\(\boxed{-150}\)[/tex].
