An arc on a circle measures $125^{\circ}$. The measure of the central angle, in radians, is within which range?
A. 0 to $\frac{\pi}{2}$ radians
B. $\frac{\pi}{2}$ in $\pi$ radians
C. $\pi$ to $\frac{3 \pi}{2}$ radians
D. $\frac{3 \pi}{2}$ to $2 \pi$ radians
Answer (2)
Convert the given angle from degrees to radians using the conversion factor 18 0 ∘ π , resulting in 36 25 π radians.
Approximate the value of 36 25 π to be approximately 2.18 radians.
Compare the calculated value with the given ranges in radians.
Determine that the central angle falls within the range 2 π to π radians. 2 π to π radians
Explanation
Problem Analysis We are given an arc on a circle that measures 12 5 ∘ . We need to find the range in radians that contains the central angle corresponding to this arc.
Convert Degrees to Radians First, we need to convert the angle from degrees to radians. To do this, we use the conversion factor 18 0 ∘ π .
Calculate Central Angle in Radians The central angle in radians is calculated as follows: 12 5 ∘ × 18 0 ∘ π = 180 125 π = 36 25 π ≈ 2.18 radians
Determine the Range Now, we need to determine which of the given ranges contains the value 36 25 π ≈ 2.18 radians.
Let's examine the given ranges:
0 to 2 π radians (approximately 0 to 1.57 radians)
2 π to π radians (approximately 1.57 to 3.14 radians)
π to 2 3 π radians (approximately 3.14 to 4.71 radians)
2 3 π to 2 π radians (approximately 4.71 to 6.28 radians)
Final Answer Comparing the calculated central angle of approximately 2.18 radians with the given ranges, we see that it falls within the range of 2 π to π radians, since 1.57 < 2.18 < 3.14 .
Conclusion Therefore, the measure of the central angle, in radians, is within the range of 2 π to π radians.
Examples
Understanding angles in radians is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a pendulum, the angle of displacement is often measured in radians. Similarly, in electrical engineering, the phase difference between two alternating current signals is expressed in radians. Knowing how to convert between degrees and radians and understanding the magnitude of angles in radians helps in solving practical problems in these fields.
The measure of the central angle corresponding to an arc of 12 5 ∘ is 36 25 π radians, which approximates to 2.18 radians. This value falls within the range of 2 π to p i radians. Therefore, the correct option is B.
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