To determine the value of [tex]\(\sin 60^\circ\)[/tex], let's follow these steps:
1. Understanding sine of common angles:
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 certain special angles like [tex]\(30^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(60^\circ\)[/tex], the values of sine are well known.
2. Using the property of 60 degrees:
For a [tex]\(60^\circ\)[/tex] angle, there is a well-known result in trigonometry. Specifically, in an equilateral triangle, if we split it in half, we create a right triangle with angles [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex]. For this triangle, the ratio of the sides are as follows:
- Hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle): 1 (in terms of unit length)
- Opposite to [tex]\(60^\circ\)[/tex]: [tex]\(\sqrt{3}/2\)[/tex] of the hypotenuse
- Opposite to [tex]\(30^\circ\)[/tex]: [tex]\(1/2\)[/tex] of the hypotenuse
3. Recognizing the sine value for [tex]\(60^\circ\)[/tex]:
From the above properties:
[tex]\[
\sin 60^\circ = \frac{\text{opposite side to } 60^\circ}{\text{hypotenuse}} = \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}
\][/tex]
4. Comparing against given options:
Let's match this value to the provided choices:
- A. [tex]\(\frac{1}{2}\)[/tex]
- B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- C. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
- D. 1
- E. [tex]\(\sqrt{3}\)[/tex]
- F. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
5. Conclusion:
The value [tex]\(\frac{\sqrt{3}}{2}\)[/tex] corresponds to option F in the given choices.
Thus, the correct answer is:
[tex]\[
\boxed{F}
\][/tex]
