Let's solve the given trigonometric equation step-by-step.
1. Understand the given equation:
[tex]\[\sin \left(90^{\circ} - x\right) = -\frac{\sqrt{3}}{2}\][/tex]
2. Use the co-function identity for sine:
Recall that [tex]\(\sin \left(90^{\circ} - x\right) = \cos(x)\)[/tex]. This allows us to rewrite the given equation in terms of cosine:
[tex]\[\cos(x) = -\frac{\sqrt{3}}{2}\][/tex]
3. Determine the angles where [tex]\(\cos(x) = -\frac{\sqrt{3}}{2}\)[/tex]:
The values of [tex]\(x\)[/tex] for which [tex]\(\cos(x) = -\frac{\sqrt{3}}{2}\)[/tex] are known from the unit circle. These angles are:
- [tex]\( 150^{\circ} \)[/tex] (since [tex]\( \cos(150^{\circ}) = -\frac{\sqrt{3}}{2} \)[/tex])
- [tex]\( 210^{\circ} \)[/tex] (since [tex]\( \cos(210^{\circ}) = -\frac{\sqrt{3}}{2} \)[/tex])
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation is:
[tex]\[
\boxed{150^{\circ}\ \text{and}\ 210^{\circ}}
\][/tex]
