To understand the input and output values for determining the sine of [tex]\(60^\circ\)[/tex], let's analyze the trigonometric functions and conversions:
1. Understanding Sine Function:
- The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
- For specific standard angles, the values of sine, cosine, etc., are well-known.
2. Calculating [tex]\(\sin(60^\circ)\)[/tex]:
- The value of [tex]\(\sin(60^\circ)\)[/tex] is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
3. Conversion between Degrees and Radians:
- Angles can be represented in degrees or radians. To convert degrees to radians, we use the formula:
[tex]\[
\text{radians} = \left(\frac{\pi}{180}\right) \times \text{degrees}
\][/tex]
- For example, [tex]\(60^\circ = \left(\frac{\pi}{180}\right) \times 60 = \frac{\pi}{3} \)[/tex] radians.
4. Finding Inputs and Outputs:
Let’s examine the four provided input and output pairs:
### Pair 1:
- Input: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
- Output: [tex]\(60^\circ\)[/tex]
Explanation:
To find what angle has the sine value of [tex]\(\frac{2}{\sqrt{3}}\)[/tex], we calculate:
[tex]\[
\sin^{-1}\left(\frac{2}{\sqrt{3}}\right)
\][/tex]
However, simplifying [tex]\(\frac{2}{\sqrt{3}}\)[/tex]:
[tex]\[
\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\][/tex]
Since [tex]\(\frac{2\sqrt{3}}{3}\)[/tex] is not a standard sine value and does not match [tex]\(60^\circ\)[/tex]'s sine value ([tex]\(\frac{\sqrt{3}}{2}\)[/tex]), this pair does not seem correct by observation.
### Pair 2:
- Input: [tex]\(60^\circ\)[/tex]
- Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
Explanation:
- Here, we know:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, this pair is correct as [tex]\(\sin(60^\circ)\)[/tex] is indeed [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
### Pair 3:
- Input: [tex]\(60^\circ\)[/tex]
- Output: [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
Explanation:
As explained earlier, the sine of [tex]\(60^\circ\)[/tex] is [tex]\(\frac{\sqrt{3}}{2}\)[/tex]. Since there’s no trigonometric function directly giving an output of [tex]\(\frac{2}{\sqrt{3}}\)[/tex] yet corresponds to [tex]\(60^\circ\)[/tex], this pair is incorrect.
### Pair 4:
- Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Output: [tex]\(60^\circ\)[/tex]
Explanation:
We need to find the angle for which the sine is [tex]\(\frac{\sqrt{3}}{2}\)[/tex]:
[tex]\[
\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = 60^\circ
\][/tex]
This pair is correct as [tex]\( \frac{\sqrt{3}}{2} \)[/tex] is the sine value for [tex]\( 60^\circ \)[/tex].
### Conclusion:
Combining our findings, we confirm pairs:
1. Input: [tex]\(60^\circ\)[/tex]; Output: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
2. Input: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]; Output: [tex]\(60^\circ\)[/tex]
These adhere to the known trigonometric values for [tex]\(60^\circ\)[/tex].
