Let's clarify the given input and output values related to determining the sine of [tex]\( 60^\circ \)[/tex].
1. Input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex], Output: [tex]\(60^\circ\)[/tex]
This does not correspond to standard calculations involving the sine of [tex]\( 60^\circ \)[/tex]. The fraction [tex]\(\frac{2}{\sqrt{3}}\)[/tex] does not directly relate to the sine of [tex]\( 60^\circ \)[/tex], so this input-output pair is incorrect.
2. Input: [tex]\( 60^\circ\)[/tex], Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
This is correct because the sine of [tex]\( 60^\circ \)[/tex] is [tex]\(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)[/tex].
3. Input: [tex]\(60^\circ\)[/tex], Output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
Again, this does not correspond to the standard trigonometric function values for [tex]\( 60^\circ \)[/tex]. Thus, this input-output pair is incorrect.
4. Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex], Output: [tex]\( 60^\circ\)[/tex]
This is correct. The angle whose sine is [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is [tex]\( 60^\circ \)[/tex]. Hence, [tex]\(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = 60^\circ\)[/tex].
Summarizing the correct pairs:
- Input: [tex]\(60^\circ\)[/tex], Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex], Output: [tex]\(60^\circ\)[/tex]
To conclude, the correct input-output values for determining the sine of [tex]\( 60^\circ \)[/tex] are:
- When the input is [tex]\(60^\circ\)[/tex], the output is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
- When the input is [tex]\(\frac{\sqrt{3}}{2}\)[/tex], the output is [tex]\(60^\circ\)[/tex].
