To determine [tex]\(\sin 45^{\circ}\)[/tex], we can use some fundamental properties of trigonometry.
First, we know that [tex]\(45^\circ\)[/tex] is an angle in a right triangle where both legs are of equal length. Given this is a right-angle isosceles triangle, we can use the unit circle or the special 45°-45°-90° triangle properties.
In a [tex]\(45^\circ\)[/tex] - [tex]\(45^\circ\)[/tex] - [tex]\(90^\circ\)[/tex] triangle, the ratio of the lengths of the sides opposite the [tex]\(45^\circ\)[/tex] angles is 1:1, meaning the legs are equal. The length of the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of one of the legs.
Therefore, the sine of [tex]\(45^\circ\)[/tex] is given by the ratio of the length of the opposite side to the hypotenuse.
[tex]\[
\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}
\][/tex]
To express [tex]\(\frac{1}{\sqrt{2}}\)[/tex] in a more standard form, we multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
In decimal form, [tex]\(\frac{\sqrt{2}}{2} \)[/tex] is approximately equal to 0.7071067811865475. Given this, we see that the sine of [tex]\(45^\circ\)[/tex] is:
[tex]\[
\sin 45^\circ \approx 0.7071067811865475
\][/tex]
Among the answer choices provided:
[tex]\[
A. \frac{1}{\sqrt{2}}
B. 1
C. \frac{1}{2}
D. \sqrt{2}
\][/tex]
The correct answer is [tex]\(A. \frac{1}{\sqrt{2}}\)[/tex], which is the standard trigonometric value expressed for [tex]\(\sin 45^\circ\)[/tex].
