An arc on a circle measures $8^{\circ}$. The measure of the central angle, in radians, is within which range?
A. 0 to $\frac{\pi}{2}$ radians
B. $\frac{\pi}{2}$ to $\pi$ radians
C. $\pi$ to $\frac{3 \pi}{2}$ radians
D. $\frac{3 \pi}{2}$ to $2\pi$ radians
Answer (1)
Convert the angle from degrees to radians using the conversion factor: 45 2 π .
Approximate the value of 45 2 π : ≈ 0.1396 .
Compare the approximated value with the given ranges.
Determine that the radian measure is within the range: 0 to 2 π radians .
Explanation
Problem Analysis We are given an arc on a circle that measures 8 ∘ . We need to find the range in which the measure of the central angle in radians lies. To convert from degrees to radians, we use the conversion factor 18 0 ∘ π .
Convert to Radians First, convert the angle from degrees to radians: 8 ∘ × 18 0 ∘ π = 180 8 π = 45 2 π radians Now, we need to determine which range the value 45 2 π falls within.
Approximate and Compare We know that π ≈ 3.14159 . Therefore, 45 2 π ≈ 45 2 × 3.14159 ≈ 45 6.28318 ≈ 0.1396 Now we compare this value to the given ranges:
0 to 2 π radians: 2 π ≈ 2 3.14159 ≈ 1.5708 . Since 0.1396 is between 0 and 1.5708 , this is a possible range.
2 π to π radians: This range is approximately 1.5708 to 3.14159 . Since 0.1396 is not in this range, this is not the correct range.
π to 2 3 π radians: This range is approximately 3.14159 to 4.7124 . Since 0.1396 is not in this range, this is not the correct range.
2 3 π to 2 π radians: This range is approximately 4.7124 to 6.2832 . Since 0.1396 is not in this range, this is not the correct range.
Final Answer Since 45 2 π ≈ 0.1396 and 2 π ≈ 1.5708 , we have 0 < 45 2 π < 2 π . Therefore, the measure of the central angle in radians is within the range of 0 to 2 π radians.
Examples
Understanding how to convert between degrees and radians is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a pendulum, we need to express the angle of displacement in radians to use it in equations of motion. Similarly, in computer graphics, angles are often represented in radians for calculations involving rotations and transformations.
