Certainly! Let's break down how to determine the correct input and output values for finding the sine of [tex]\(60^\circ\)[/tex]:
1. Given angle (input): We start with the angle [tex]\(60^\circ\)[/tex]. This angle is our input.
2. Calculating the sine (output): We need to determine the sine of this angle, which requires evaluating [tex]\(\sin(60^\circ)\)[/tex].
From trigonometry, especially using the unit circle or standard trigonometric tables, we know the sine of [tex]\(60^\circ\)[/tex]:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
So:
- Input: [tex]\(60^\circ\)[/tex]
- Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Comparing this to the provided options:
1. input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]; output: [tex]\(60^\circ\)[/tex] – This is incorrect. The input value is not the angle given in degrees.
2. input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex] – This is correct. The angle [tex]\(60^\circ\)[/tex] yields [tex]\(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)[/tex].
3. input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex] – This is incorrect. The sine of [tex]\(60^\circ\)[/tex] is not [tex]\(\frac{2}{\sqrt{3}}\)[/tex].
4. input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; output: [tex]\(60^\circ\)[/tex] – This is incorrect. The [tex]\(\frac{\sqrt{3}}{2}\)[/tex] represents the sine value, not the angle.
Therefore, the correct input and output values are:
- Input: [tex]\(60^\circ\)[/tex]
- Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Thus, the correct selection is:
input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
