Answer :
To determine whether the statement "One radian is smaller than one degree" is true or false, we need to understand the relationship between radians and degrees.
1. By definition, one complete revolution around a circle is \(360^{\circ}\), and the same complete revolution is also equal to \(2\pi\) radians. Therefore, \(2\pi\) radians = \(360^{\circ}\).
2. To convert from radians to degrees, we can use the conversion factor:
[tex]\[ 1 \text{ radian} = \left(\frac{180}{\pi}\right)^{\circ} \][/tex]
Using this conversion factor, we can find the value of one radian in degrees:
[tex]\[ 1 \text{ radian} \approx 57.29577951308232^{\circ} \][/tex]
This calculation shows that one radian is approximately equal to \(57.29578\) degrees, which is much larger than one degree.
3. Now we analyze the given choices:
- Option A: The statement here says 1 radian = 1 degree, which is incorrect because 1 radian is approximately 57.29578 degrees, not 1 degree.
- Option B: The statement correctly identifies that 1 radian is approximately equal to \( \left(\frac{180}{\pi}\right)^{\circ} \approx 57.29578^{\circ} \). This is the correct interpretation of the conversion from radians to degrees and proves that one radian is indeed larger than one degree.
- Option C: This statement suggests that 1 radian is approximately 0.0174533 degrees, which incorrectly applies the inverse conversion and gives a very small value, thus it is incorrect.
- Option D: This option discusses the measure of a full revolution in radians as opposed to degrees, which, although true, does not directly address the comparison between one radian and one degree.
Given the accurate conversion and the correct reasoning provided in Option B, the proper choice is:
B. The statement is false because 1 radian \(= \left(\frac{180}{\pi}\right)^{\circ} \approx 57.29578^{\circ}
1. By definition, one complete revolution around a circle is \(360^{\circ}\), and the same complete revolution is also equal to \(2\pi\) radians. Therefore, \(2\pi\) radians = \(360^{\circ}\).
2. To convert from radians to degrees, we can use the conversion factor:
[tex]\[ 1 \text{ radian} = \left(\frac{180}{\pi}\right)^{\circ} \][/tex]
Using this conversion factor, we can find the value of one radian in degrees:
[tex]\[ 1 \text{ radian} \approx 57.29577951308232^{\circ} \][/tex]
This calculation shows that one radian is approximately equal to \(57.29578\) degrees, which is much larger than one degree.
3. Now we analyze the given choices:
- Option A: The statement here says 1 radian = 1 degree, which is incorrect because 1 radian is approximately 57.29578 degrees, not 1 degree.
- Option B: The statement correctly identifies that 1 radian is approximately equal to \( \left(\frac{180}{\pi}\right)^{\circ} \approx 57.29578^{\circ} \). This is the correct interpretation of the conversion from radians to degrees and proves that one radian is indeed larger than one degree.
- Option C: This statement suggests that 1 radian is approximately 0.0174533 degrees, which incorrectly applies the inverse conversion and gives a very small value, thus it is incorrect.
- Option D: This option discusses the measure of a full revolution in radians as opposed to degrees, which, although true, does not directly address the comparison between one radian and one degree.
Given the accurate conversion and the correct reasoning provided in Option B, the proper choice is:
B. The statement is false because 1 radian \(= \left(\frac{180}{\pi}\right)^{\circ} \approx 57.29578^{\circ}
