Answer :
To determine the value of [tex]\(\sin 45^\circ\)[/tex], we will proceed as follows:
First, we recognize that [tex]\(45^\circ\)[/tex] is an angle that is commonly found in the unit circle, and we can rely on properties of a special right triangle, specifically the 45-45-90 triangle. In such triangles, the two non-hypotenuse sides are equal, and the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of one of these sides.
1. Construct a 45-45-90 triangle in which each leg has length 1. The hypotenuse of this triangle will be [tex]\(\sqrt{2}\)[/tex], using the Pythagorean theorem:
[tex]\[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}. \][/tex]
2. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. For [tex]\(\sin 45^\circ\)[/tex], it is:
[tex]\[ \sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}. \][/tex]
3. This ratio [tex]\(\frac{1}{\sqrt{2}}\)[/tex] simplifies by rationalizing the denominator:
[tex]\[ \frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}. \][/tex]
However, among the provided choices, we notice that the simplified form of [tex]\(\frac{1}{\sqrt{2}}\)[/tex] which we calculated is already listed. So, we recognize our ratio doesn't need further simplification for this specific numbered list.
Considering the numerical result [tex]\(0.7071067811865475\)[/tex]:
4. Converting the numerical result to a fractional form is not necessary because one of our choices matches [tex]\(\frac{1}{\sqrt{2}}\)[/tex], which corresponds to our calculated value for [tex]\(\sin 45^\circ\)[/tex].
Thus, the correct answer is:
B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
First, we recognize that [tex]\(45^\circ\)[/tex] is an angle that is commonly found in the unit circle, and we can rely on properties of a special right triangle, specifically the 45-45-90 triangle. In such triangles, the two non-hypotenuse sides are equal, and the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of one of these sides.
1. Construct a 45-45-90 triangle in which each leg has length 1. The hypotenuse of this triangle will be [tex]\(\sqrt{2}\)[/tex], using the Pythagorean theorem:
[tex]\[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}. \][/tex]
2. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. For [tex]\(\sin 45^\circ\)[/tex], it is:
[tex]\[ \sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}. \][/tex]
3. This ratio [tex]\(\frac{1}{\sqrt{2}}\)[/tex] simplifies by rationalizing the denominator:
[tex]\[ \frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}. \][/tex]
However, among the provided choices, we notice that the simplified form of [tex]\(\frac{1}{\sqrt{2}}\)[/tex] which we calculated is already listed. So, we recognize our ratio doesn't need further simplification for this specific numbered list.
Considering the numerical result [tex]\(0.7071067811865475\)[/tex]:
4. Converting the numerical result to a fractional form is not necessary because one of our choices matches [tex]\(\frac{1}{\sqrt{2}}\)[/tex], which corresponds to our calculated value for [tex]\(\sin 45^\circ\)[/tex].
Thus, the correct answer is:
B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
