Certainly! Let's determine the correct input and output values for the sine of [tex]\(60^\circ\)[/tex].
### Step-by-Step Solution:
1. Identify the Problem:
We need to determine the sine of [tex]\(60^\circ\)[/tex].
2. Relevant Trigonometric Concept:
The sine function, denoted as [tex]\(\sin(\theta)\)[/tex], is a fundamental trigonometric function. We are specifically interested in [tex]\(\sin(60^\circ)\)[/tex].
3. Known Value of Sine for [tex]\(60^\circ\)[/tex]:
From trigonometric tables or the knowledge of special angles in trigonometry, we know that:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
4. Match with Provided Options:
- Option 1: Input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]; Output: [tex]\(60^\circ\)[/tex]
- This option does not align with the standard sine value for [tex]\(60^\circ\)[/tex].
- Option 2: Input: [tex]\(60^\circ\)[/tex]; Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- This correctly matches our determination where [tex]\(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)[/tex].
- Option 3: Input: [tex]\(60^\circ\)[/tex]; Output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
- This does not align with the known sine of [tex]\(60^\circ\)[/tex].
- Option 4: Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; Output: [tex]\(60^\circ\)[/tex]
- This is the inverse function statement where [tex]\(\sin^{-1}(\frac{\sqrt{3}}{2}) = 60^\circ\)[/tex]. While mathematically true, it does not match the format given in the question which asks for [tex]\(\sin(60^\circ)\)[/tex].
### Conclusion:
The correct input and output values for the sine of [tex]\(60^\circ\)[/tex] are:
[tex]\[
\text{Input: } 60^\circ \quad \text{and} \quad \text{Output: } \frac{\sqrt{3}}{2}
\][/tex]
Thus, [tex]\( \boxed{\text{Input: } 60^\circ; \text{ Output: } \frac{\sqrt{3}}{2}} \)[/tex] is the correct answer.
