To find the value of [tex]\(\sin 60^\circ\)[/tex], we need to recall the value of the sine function for 60 degrees. The sine of 60 degrees is a well-known value in trigonometry.
Let's consider the unit circle or the 30-60-90 special right triangle, which is a standard right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. In such a triangle, the ratios of the sides are as follows:
- The length of the side opposite the 30-degree angle is [tex]\(\frac{1}{2}\)[/tex].
- The length of the side opposite the 60-degree angle is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
- The length of the hypotenuse is 1.
In this triangle, for the 60-degree angle, the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
[tex]\[
\sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}
\][/tex]
Thus, the value of [tex]\(\sin 60^\circ\)[/tex] is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Comparing this with the given options, we see that:
- Option A: [tex]\(\frac{1}{2}\)[/tex]
- Option B: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Option C: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- Option D: [tex]\(\sqrt{3}\)[/tex]
- Option E: [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
- Option F: 1
Therefore, [tex]\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex], which corresponds to Option B.
