Certainly! Let's proceed step-by-step to address the given question, which includes both the sine function and its inverse.
### Sine Function and Its Inverse
1. Sine of [tex]\(60^{\circ}\)[/tex]
The sine function is defined for an angle [tex]\(\theta\)[/tex] as [tex]\(\sin(\theta)\)[/tex]. For [tex]\(\theta = 60^{\circ}\)[/tex]:
[tex]\[
\sin(60^{\circ}) = \frac{\sqrt{3}}{2}
\][/tex]
2. Inverse Sine (Arcsine) Function
The inverse sine function, [tex]\(\sin^{-1}(x)\)[/tex] or [tex]\(\arcsin(x)\)[/tex], gives the angle whose sine is [tex]\(x\)[/tex].
For the input [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
[tex]\[
\arcsin\left(\frac{\sqrt{3}}{2}\right) = 60^{\circ}
\][/tex]
### Evaluating the Options
Now, let's evaluate the provided options:
1. Option 1: Input [tex]\(\frac{2}{\sqrt{3}}\)[/tex]; Output [tex]\(60^{\circ}\)[/tex]
The input value [tex]\(\frac{2}{\sqrt{3}}\)[/tex] is not a standard sine value of [tex]\(60^{\circ}\)[/tex]. Thus, this option is incorrect.
2. Option 2: Input [tex]\(60^{\circ}\)[/tex]; Output [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
As we calculated above:
[tex]\[
\sin(60^{\circ}) = \frac{\sqrt{3}}{2}
\][/tex]
This option is correct.
3. Option 3: Input [tex]\(60^{\circ}\)[/tex]; Output [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
This output [tex]\(\frac{2}{\sqrt{3}}\)[/tex] does not match the actual sine value of [tex]\(60^{\circ}\)[/tex]. Thus, this option is incorrect.
4. Option 4: Input [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; Output [tex]\(60^{\circ}\)[/tex]
Using the arcsine function:
[tex]\[
\arcsin\left(\frac{\sqrt{3}}{2}\right) = 60^{\circ}
\][/tex]
This option is correct.
### Conclusion
The correct input and output pairs for determining the sine of [tex]\(60^{\circ}\)[/tex] and its inverse 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]
Thus, the correct options 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]
