To determine the possible values of [tex]\( x \)[/tex] for the point [tex]\(\left(x, \frac{\sqrt{3}}{2}\right)\)[/tex] on the unit circle, we start by using the equation of the unit circle:
[tex]\[ x^2 + y^2 = 1. \][/tex]
Given that the y-coordinate is [tex]\(\frac{\sqrt{3}}{2}\)[/tex], we substitute [tex]\( y \)[/tex] with [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
[tex]\[ x^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1. \][/tex]
Next, we calculate [tex]\(\left( \frac{\sqrt{3}}{2} \right)^2\)[/tex]:
[tex]\[ \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4}. \][/tex]
Substituting [tex]\(\frac{3}{4}\)[/tex] in the unit circle equation:
[tex]\[ x^2 + \frac{3}{4} = 1. \][/tex]
Now, we solve for [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = 1 - \frac{3}{4}. \][/tex]
[tex]\[ x^2 = \frac{4}{4} - \frac{3}{4}. \][/tex]
[tex]\[ x^2 = \frac{1}{4}. \][/tex]
To find [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \frac{1}{2} \text{ or } x = -\frac{1}{2}. \][/tex]
Therefore, the possible values of [tex]\( x \)[/tex] are:
[tex]\[ x = \frac{1}{2} \text{ or } x = -\frac{1}{2}. \][/tex]
Given the answer choices:
A. [tex]\(-\frac{\sqrt{3}}{2}\)[/tex]
B. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
C. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
D. [tex]\(\frac{1}{2}\)[/tex]
Only option D, [tex]\(\frac{1}{2}\)[/tex], matches the possible values of [tex]\( x \)[/tex] that we calculated.
Hence, the correct answer is:
[tex]\[ \boxed{\frac{1}{2}}. \][/tex]
