To find the value of [tex]\(\cot(30^{\circ})\)[/tex] given that [tex]\(\sin(30^{\circ}) = \frac{1}{2}\)[/tex] and [tex]\(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex], we can use the trigonometric identity for cotangent:
[tex]\[
\cot(x) = \frac{\cos(x)}{\sin(x)}
\][/tex]
Let's apply this identity to [tex]\(30^{\circ}\)[/tex]:
1. Write the cotangent identity:
[tex]\[
\cot(30^{\circ}) = \frac{\cos(30^{\circ})}{\sin(30^{\circ})}
\][/tex]
2. Substitute the known values of [tex]\(\sin(30^{\circ})\)[/tex] and [tex]\(\cos(30^{\circ})\)[/tex]:
[tex]\[
\cot(30^{\circ}) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}
\][/tex]
3. Simplify the fraction:
[tex]\[
\cot(30^{\circ}) = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}
\][/tex]
Therefore, the value of [tex]\(\cot(30^{\circ})\)[/tex] is:
[tex]\[
\cot(30^{\circ}) = \sqrt{3}
\][/tex]
Given the provided options are:
[tex]\[
\frac{1}{2}
\][/tex]
[tex]\[
\frac{\sqrt{3}}{3}
\][/tex]
[tex]\[
\frac{\sqrt{3}}{2}
\][/tex]
None of these correspond directly to [tex]\(\sqrt{3}\)[/tex]. Considering only the integer part of [tex]\(\sqrt{3}\approx 1.732\)[/tex] among the options:
[tex]\(\cot(30^{\circ})\)[/tex] simplifies exactly to one of the provided options. In this context, the closest integer match calculating it precisely is [tex]\(\approx 1.7320508075688772\)[/tex]. Therefore, none matches strictly with specific option list.
The cotangent value determined demonstrates the closest precise value indeed provided is :
[tex]\(\boxed{\sqrt{3}}.\)[/tex]
Given exact numerical understanding:
The value correctly matched is indeed approximating [tex]\(1.7320508075688772\)[/tex].
