Certainly! Let's solve the equation [tex]\(\sqrt{3} \cot(x) - 1 = -2\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex].
### Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
\sqrt{3} \cot(x) - 1 = -2
\][/tex]
2. Isolate [tex]\(\cot(x)\)[/tex]:
Add 1 to both sides of the equation:
[tex]\[
\sqrt{3} \cot(x) = -1 + 1
\][/tex]
Simplify the right-hand side:
[tex]\[
\sqrt{3} \cot(x) = -1 + 1 = -1 + 1 = 0
\][/tex]
3. Divide both sides by [tex]\(\sqrt{3}\)[/tex]:
To isolate [tex]\(\cot(x)\)[/tex]:
[tex]\[
\cot(x) = -\frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}}
\][/tex]
4. Recognize the exact value:
We know that:
[tex]\[
\cot(x) = \frac{\cos(x)}{\sin(x)} = -\frac{1}{\sqrt{3}}
\][/tex]
This implies:
[tex]\[
\frac{\cos(x)}{\sin(x)} = -\frac{1}{\sqrt{3}}
\][/tex]
5. Find the corresponding angle:
The cotangent function [tex]\(\cot(x) = -\frac{1}{\sqrt{3}}\)[/tex] corresponds to angles where the tangent function [tex]\(\tan(x) = -\sqrt{3}\)[/tex] because [tex]\(\cot(x) = \frac{1}{\tan(x)}\)[/tex].
The value [tex]\(\tan(x) = -\sqrt{3}\)[/tex] occurs at:
[tex]\[
x = \frac{4\pi}{3}, \quad x = \frac{7\pi}{3}
\][/tex]
6. Verify these solutions within the interval [tex]\([0, 2\pi)\)[/tex]:
[tex]\[
\frac{4\pi}{3} \quad \text{is in} \quad \left[0, 2\pi\right)
\][/tex]
[tex]\[
\frac{7\pi}{3} \quad \text{is not in} \quad \left[0, 2\pi\right)
\][/tex]
Therefore, within the interval [tex]\([0, 2\pi)\)[/tex], the valid solution is:
[tex]\[
x = \frac{4\pi}{3}
\][/tex]
Thus, the solution to the equation [tex]\(\sqrt{3} \cot(x) - 1 = -2\)[/tex] within the interval [tex]\([0, 2 \pi)\)[/tex] is:
[tex]\[
\boxed{\frac{4\pi}{3}}
\][/tex]
