To solve for [tex]\( x \)[/tex] that satisfies the equation [tex]\(\cot \left(90^{\circ} - x\right) = -\frac{\sqrt{3}}{3}\)[/tex], let's follow these steps:
1. Recall the cotangent identity:
[tex]\[
\cot(90^\circ - x) = \tan(x)
\][/tex]
This transformation uses the co-function identity for cotangent.
2. Rewrite the original equation:
[tex]\[
\tan(x) = -\frac{\sqrt{3}}{3}
\][/tex]
3. Find the reference angle for [tex]\(\tan(x) = -\frac{\sqrt{3}}{3}\)[/tex]:
We know that [tex]\(\tan(30^\circ) = \frac{\sqrt{3}}{3}\)[/tex]. The negative tangent values occur in the second and fourth quadrants. Therefore, we need to find angles in these quadrants.
4. Determine the angles in relevant quadrants:
In the second quadrant, the corresponding angle where [tex]\(\tan(x)\)[/tex] is negative and equals [tex]\(-\frac{\sqrt{3}}{3}\)[/tex] is:
[tex]\[
180^\circ - 30^\circ = 150^\circ
\][/tex]
Checking the other quadrants, specifically the fourth, would give us an angle, but given the choices, let's ensure we consider the closest match.
5. Verify the solution against the provided answer choices:
Let's review the given options:
- A. [tex]\(120^\circ\)[/tex]
- B. [tex]\(240^\circ\)[/tex]
- C. [tex]\(210^\circ\)[/tex]
- D. [tex]\(150^\circ\)[/tex]
Comparing 150° directly to the options given, it matches [tex]\(150^\circ\)[/tex].
Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\cot\left(90^\circ - x\right) = -\frac{\sqrt{3}}{3}\)[/tex] is:
[tex]\[
\boxed{150^\circ}
\][/tex]
