Connect with a global community of knowledgeable individuals on IDNLearn.com. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.
Sagot :
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].
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].
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for choosing IDNLearn.com for your queries. We’re committed to providing accurate answers, so visit us again soon.
