To determine the sine of [tex]\(60^\circ\)[/tex] and find the corresponding input and output values, we need to follow a step-by-step approach by considering the properties of the sine function and the well-known values for specific angles.
1. Understanding Sine of [tex]\(60^\circ\)[/tex]:
- The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse.
- For the angle [tex]\(60^\circ\)[/tex], which is a common angle whose sine value is well-known, we have:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
So, when the input is the angle [tex]\(60^\circ\)[/tex], the output, which is the value of sine for that angle, is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
- Input: [tex]\(60^\circ\)[/tex]
- Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
2. Determining the Angle for a Given Sine Value:
- When we have a given value for sine and want to find the corresponding angle, we look for the angle whose sine gives that value.
- In this case, if the sine value is [tex]\(\frac{\sqrt{3}}{2}\)[/tex], the angle whose sine corresponds to this value is [tex]\(60^\circ\)[/tex].
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Output: [tex]\(60^\circ\)[/tex]
Given the above information, let's summarize the input and output values:
First scenario:
- Input: [tex]\(60^\circ\)[/tex]
- Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Second scenario:
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Output: [tex]\(60^\circ\)[/tex]
Comparing this with the given multiple choices:
input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]; output: [tex]\(60^\circ\)[/tex] – This is incorrect because [tex]\(\frac{2}{\sqrt{3}}\)[/tex] is not the sine value of [tex]\(60^\circ\)[/tex].
input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex] – This is correct.
input: [tex]\(60^\circ\)[/tex]; output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex] – This is incorrect because the output should be [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; output: [tex]\(60^\circ\)[/tex] – This is correct.
Thus, we have the correct pairings:
- Input: [tex]\(60^\circ\)[/tex]; Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; Output: [tex]\(60^\circ\)[/tex]
Therefore, the correct answers are the second and the fourth options.
