To find the value of [tex]\(\sin 45^{\circ}\)[/tex], we start by considering the properties of an angle of [tex]\(45^{\circ}\)[/tex].
The angle [tex]\(45^{\circ}\)[/tex] is a special angle in trigonometry, which is found in right-angled triangles and also in the unit circle. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
A [tex]\(45^{\circ}\)[/tex] angle is found in an isosceles right triangle, where the two legs are of equal length. This triangle essentially splits a square in half diagonally. If each leg of the isosceles right triangle is of length [tex]\(1\)[/tex], then we can calculate the length of the hypotenuse using the Pythagorean theorem:
[tex]\[
\text{hypotenuse}^2 = 1^2 + 1^2 = 2
\][/tex]
This gives:
[tex]\[
\text{hypotenuse} = \sqrt{2}
\][/tex]
Now, the sine of [tex]\(45^{\circ}\)[/tex] is:
[tex]\[
\sin 45^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
\][/tex]
To rationalize the denominator, we multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\sin 45^{\circ} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
The numerical value of [tex]\(\frac{1}{\sqrt{2}}\)[/tex] or equivalently [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is approximately [tex]\(0.7071067811865475\)[/tex], which confirms our result.
Therefore, the correct value for [tex]\(\sin 45^{\circ}\)[/tex] among the provided options is:
[tex]\[
\boxed{\frac{1}{\sqrt{2}}}
\][/tex]
