Answer :
To find the value of [tex]\(\sin 60^{\circ}\)[/tex], we can recall the exact trigonometric values for common angles. The angle [tex]\(60^\circ\)[/tex] is one of these common angles, and its sine value is well-known from trigonometric tables or the unit circle.
### Step-by-Step Solution:
1. Recall the special triangles: One way to determine [tex]\(\sin 60^{\circ}\)[/tex] is by considering a 30-60-90 triangle. In such a triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex], where:
- The side opposite [tex]\(30^\circ\)[/tex] is [tex]\(1\)[/tex].
- The side opposite [tex]\(60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2\)[/tex].
2. Define [tex]\(\sin\)[/tex] for a given angle: The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
For [tex]\(60^\circ\)[/tex]:
[tex]\[ \sin 60^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \][/tex]
3. Identify the correct option: According to the given options:
- [tex]\( \sqrt{3} \)[/tex]
- [tex]\( \frac{1}{2} \)[/tex]
- [tex]\( \frac{1}{\sqrt{3}} \)[/tex]
- [tex]\( \frac{1}{\sqrt{2}} \)[/tex]
- [tex]\( 1 \)[/tex]
- [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
The option that represents [tex]\( \frac{\sqrt{3}}{2} \)[/tex] is [tex]\( \boxed{F} \)[/tex].
Thus, [tex]\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)[/tex], and the correct option is [tex]\( \boxed{F} \)[/tex].
### Step-by-Step Solution:
1. Recall the special triangles: One way to determine [tex]\(\sin 60^{\circ}\)[/tex] is by considering a 30-60-90 triangle. In such a triangle, the sides are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex], where:
- The side opposite [tex]\(30^\circ\)[/tex] is [tex]\(1\)[/tex].
- The side opposite [tex]\(60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2\)[/tex].
2. Define [tex]\(\sin\)[/tex] for a given angle: The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
For [tex]\(60^\circ\)[/tex]:
[tex]\[ \sin 60^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \][/tex]
3. Identify the correct option: According to the given options:
- [tex]\( \sqrt{3} \)[/tex]
- [tex]\( \frac{1}{2} \)[/tex]
- [tex]\( \frac{1}{\sqrt{3}} \)[/tex]
- [tex]\( \frac{1}{\sqrt{2}} \)[/tex]
- [tex]\( 1 \)[/tex]
- [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
The option that represents [tex]\( \frac{\sqrt{3}}{2} \)[/tex] is [tex]\( \boxed{F} \)[/tex].
Thus, [tex]\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)[/tex], and the correct option is [tex]\( \boxed{F} \)[/tex].
