To determine the sine of a [tex]\(60^\circ\)[/tex] angle and find the corresponding input and output values, let's go through the problem step-by-step.
### Understanding the Problem:
When dealing with trigonometric functions, specifically the sine function (sin), if we are given an angle in degrees, the sine function will provide a ratio as the result. Conversely, given a ratio, the inverse sine function ([tex]\(\sin^{-1}\)[/tex] or arcsin) will return the corresponding angle in degrees.
### Input and Output Values:
1. Given an angle and finding its sine value:
- Input: [tex]\(60^\circ\)[/tex]
- Output: The sine value of [tex]\(60^\circ\)[/tex]
The sine of [tex]\(60^\circ\)[/tex] is well-known and is given by:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
2. Given a sine value and finding the corresponding angle:
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Output: The angle whose sine value is [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
The angle that corresponds to this sine value is [tex]\(60^\circ\)[/tex].
### Checking Provided Options:
1. Option: input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]; output: [tex]\(60^\circ\)[/tex]
- [tex]\(\frac{2}{\sqrt{3}}\)[/tex] is not the correct sine value for [tex]\(60^\circ\)[/tex]. This option is incorrect.
2. Option: input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Given the angle [tex]\(60^\circ\)[/tex], the sine value is indeed [tex]\(\frac{\sqrt{3}}{2}\)[/tex]. This option is correct.
3. Option: input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
- This is incorrect because the sine of [tex]\(60^\circ\)[/tex] is not [tex]\(\frac{2}{\sqrt{3}}\)[/tex].
4. Option: input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; output: [tex]\(60^\circ\)[/tex]
- Given the sine value of [tex]\(\frac{\sqrt{3}}{2}\)[/tex], the corresponding angle is [tex]\(60^\circ\)[/tex]. This option is correct.
### Conclusion:
The correct input and output pairs for determining the sine of [tex]\(60^\circ\)[/tex] are:
- Input: [tex]\(60^\circ\)[/tex]; Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; Output: [tex]\(60^\circ\)[/tex]
These values accurately represent the relationship between the angle [tex]\(60^\circ\)[/tex] and its sine value [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
