Select the correct answer.

If the point [tex]\left(x, \frac{\sqrt{3}}{2}\right)[/tex] is on the unit circle, what could the value of [tex]x[/tex] be?

A. [tex]\frac{2}{\sqrt{3}}[/tex]

B. [tex]\frac{\sqrt{3}}{2}[/tex]

C. [tex]-\frac{\sqrt{3}}{2}[/tex]

D. [tex]\frac{1}{2}[/tex]



Answer :

Certainly! Let's solve the problem step-by-step.

Given the point [tex]\((x, \frac{\sqrt{3}}{2})\)[/tex] is on the unit circle, we need to find the possible value(s) of [tex]\(x\)[/tex].

1. Unit Circle Equation:
[tex]\[ x^2 + y^2 = 1 \][/tex]
Here, [tex]\(y = \frac{\sqrt{3}}{2}\)[/tex].

2. Substitute [tex]\(y\)[/tex] into the equation:
[tex]\[ x^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1 \][/tex]

3. Calculate [tex]\(\left(\frac{\sqrt{3}}{2}\right)^2\)[/tex]:
[tex]\[ \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{(\sqrt{3})^2}{2^2} = \frac{3}{4} \][/tex]

4. Substitute [tex]\(\frac{3}{4}\)[/tex] back into the equation:
[tex]\[ x^2 + \frac{3}{4} = 1 \][/tex]

5. Solve for [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = 1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4} \][/tex]

6. Solve for [tex]\(x\)[/tex]:
Since [tex]\(x^2 = \frac{1}{4}\)[/tex],
[tex]\[ x = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2} \][/tex]

Thus, the possible values for [tex]\(x\)[/tex] are [tex]\(\frac{1}{2}\)[/tex] and [tex]\(-\frac{1}{2}\)[/tex].

Now, let's evaluate the given options:
- A. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]: This is incorrect because [tex]\(\frac{2}{\sqrt{3}}\)[/tex] does not equate to [tex]\(\frac{1}{2}\)[/tex] or [tex]\(-\frac{1}{2}\)[/tex].
- B. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]: This is incorrect since it doesn't match [tex]\(\frac{1}{2}\)[/tex] or [tex]\(-\frac{1}{2}\)[/tex], and would not satisfy the original equation [tex]\(x^2 + y^2 = 1\)[/tex].
- C. [tex]\(-\frac{\sqrt{3}}{2}\)[/tex]: Similarly, this is incorrect for not equating [tex]\(\frac{1}{2}\)[/tex] or [tex]\(-\frac{1}{2}\)[/tex], and does not fit the unit circle criteria with the given [tex]\(y\)[/tex].
- D. [tex]\(\frac{1}{2}\)[/tex]: This is correct since [tex]\(\frac{1}{2}\)[/tex] matches one of the solutions we found.

Therefore, the correct option is:
[tex]\[ \boxed{\frac{1}{2}} \][/tex]