To solve the equation [tex]\(\sin(90^{\circ} - x) = -\frac{\sqrt{3}}{2}\)[/tex], let's proceed step-by-step:
1. Understand the Sine Complementarity:
The identity [tex]\(\sin(90^\circ - x) = \cos(x)\)[/tex] tells us that:
[tex]\[
\sin(90^\circ - x) = \cos(x)
\][/tex]
Therefore, we can rewrite the given equation as:
[tex]\[
\cos(x) = -\frac{\sqrt{3}}{2}
\][/tex]
2. Determine the Reference Angle:
The cosine value of [tex]\(\frac{\sqrt{3}}{2}\)[/tex] occurs at 30 degrees (or [tex]\(\pi/6\)[/tex] radians). However, since the cosine is negative, [tex]\(x\)[/tex] lies in the second or third quadrant.
3. Find the Possible Angles in Each Quadrant:
- In the second quadrant, the angle is [tex]\(180^\circ - 30^\circ\)[/tex]:
[tex]\[
x_1 = 180^\circ - 30^\circ = 150^\circ
\][/tex]
- In the third quadrant, the angle is [tex]\(180^\circ + 30^\circ\)[/tex]:
[tex]\[
x_2 = 180^\circ + 30^\circ = 210^\circ
\][/tex]
4. Solution:
Therefore, the values of [tex]\(x\)[/tex] that satisfy the equation are:
[tex]\[
x = 150^\circ \text{ or } 210^\circ
\][/tex]
Thus, the correct value(s) of [tex]\(x\)[/tex] that satisfy the equation are [tex]\(\boxed{150}\)[/tex] and [tex]\(\boxed{210}\)[/tex].
