To determine the input and output values for the sine of [tex]\( 60^{\circ} \)[/tex], we need to understand the standard trigonometric values.
1. Understanding the Input and Output:
- The input is the angle for which we want to find the sine value.
- The output is the sine of that input angle.
2. The Sine of [tex]\( 60^{\circ} \)[/tex]:
- The sine of [tex]\( 60^{\circ} \)[/tex] is a well-known trigonometric value.
3. Standard Value:
- From trigonometric tables or the unit circle, we know that [tex]\(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex].
4. Matching with Given Options:
- Given the options:
1. input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]; output: [tex]\(60^{\circ}\)[/tex]
2. input: [tex]\(60^{\circ}\)[/tex]; output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
3. input: [tex]\(60^{\circ}\)[/tex]; output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
4. input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; output: [tex]\(60^{\circ}\)[/tex]
5. Correct Pairing:
- The correct input-output pair matches the input angle with its sine value. Therefore, the correct input should be [tex]\(60^{\circ}\)[/tex], and the corresponding output should be [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
6. Conclusion:
- The correct pair is:
- input: [tex]\(60^{\circ}\)[/tex]; output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Hence, the input and output values for determining the sine of [tex]\( 60^{\circ} \)[/tex] are:
[tex]\[ \boxed{\text{input: } 60^{\circ}, \text{ output: } \frac{\sqrt{3}}{2}} \][/tex]
