An arc on a circle measures $295^{\circ}$. 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 18 0 ∘ π .
Calculate the radian measure: 29 5 ∘ × 18 0 ∘ π = 36 59 π ≈ 5.1487 .
Compare the calculated radian measure to the given ranges.
The central angle in radians is within the range 2 3 π to 2 π radians .
Explanation
Problem Analysis We are given an arc on a circle that measures 29 5 ∘ . We need to find the range in radians that contains the central angle.
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 Radian Measure Now, we multiply the angle in degrees by the conversion factor: 29 5 ∘ × 18 0 ∘ π = 180 295 π = 36 59 π
Approximate Radian Value Now, let's approximate the value of 36 59 π :
36 59 π ≈ 36 59 × 3.14159 ≈ 36 185.354 ≈ 5.1487
Determine the Range Now, we need to determine which of the given ranges contains this value:
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)
Since 5.1487 is between 4.71 and 6.28, the central angle in radians is within the range 2 3 π to 2 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 currents is expressed in radians. Knowing how to convert between degrees and radians allows for accurate calculations and predictions in these applications.
