Answer :
To determine the sine of \( 60^\circ \), we need to refer to the well-known values of trigonometric functions for specific angles.
### Step-by-Step Solution:
1. Understanding the unit circle and special angles:
- In trigonometry, there are certain angles which have well-known sine values. These include \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), and \( 90^\circ \).
2. Using special triangles:
- One of the commonly used triangles to find these values is an equilateral triangle, which by definition has all its angles equal to \( 60^\circ \).
- By cutting an equilateral triangle in half, we create a right triangle with angles \( 30^\circ \), \( 60^\circ \), and \( 90^\circ \).
- In such a triangle, if the sides of the equilateral triangle are of length 2, the resulting right triangle has the following properties:
- The hypotenuse (original side of the equilateral triangle) remains 2.
- The side opposite the \( 30^\circ \) angle is 1 (half of the base of the equilateral triangle).
- The side opposite the \( 60^\circ \) angle (our height) is \( \sqrt{3} \).
3. Calculating \( \sin 60^\circ \) using the right triangle:
- Sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
- For \( 60^\circ \), the side opposite is \( \sqrt{3} \) and the hypotenuse is 2.
[tex]\[ \sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \][/tex]
### Conclusion:
Based on our findings, the value of \( \sin 60^\circ \) is \(\frac{\sqrt{3}}{2}\).
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{\sqrt{3}}{2}} \][/tex]
### Step-by-Step Solution:
1. Understanding the unit circle and special angles:
- In trigonometry, there are certain angles which have well-known sine values. These include \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), and \( 90^\circ \).
2. Using special triangles:
- One of the commonly used triangles to find these values is an equilateral triangle, which by definition has all its angles equal to \( 60^\circ \).
- By cutting an equilateral triangle in half, we create a right triangle with angles \( 30^\circ \), \( 60^\circ \), and \( 90^\circ \).
- In such a triangle, if the sides of the equilateral triangle are of length 2, the resulting right triangle has the following properties:
- The hypotenuse (original side of the equilateral triangle) remains 2.
- The side opposite the \( 30^\circ \) angle is 1 (half of the base of the equilateral triangle).
- The side opposite the \( 60^\circ \) angle (our height) is \( \sqrt{3} \).
3. Calculating \( \sin 60^\circ \) using the right triangle:
- Sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
- For \( 60^\circ \), the side opposite is \( \sqrt{3} \) and the hypotenuse is 2.
[tex]\[ \sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \][/tex]
### Conclusion:
Based on our findings, the value of \( \sin 60^\circ \) is \(\frac{\sqrt{3}}{2}\).
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{\sqrt{3}}{2}} \][/tex]
