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 given 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.
Identify the correct range: 2 3 π to 2 π radians .
Explanation
Convert degrees to radians. We are given an angle of 29 5 ∘ and need to convert it to radians to determine which of the given ranges it falls within. To convert from degrees to radians, we multiply by 18 0 ∘ π .
Calculate the radian value. Let's convert 29 5 ∘ to radians: 29 5 ∘ × 18 0 ∘ π = 180 295 π = 36 59 π Now, we need to determine the approximate value of this expression to see which range it falls into.
Approximate the value. We can approximate π as 3.14159. Therefore, 36 59 × 3.14159 ≈ 36 185.35381 ≈ 5.1487
Determine the range. Now 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)
Since 5.1487 falls between 4.71 and 6.28, the correct range is 2 3 π to 2 π radians.
Examples
Understanding radian measures is crucial in many fields, such as physics and engineering, where angles are often expressed in radians for calculations involving circular motion or wave phenomena. For example, when designing a rotating machine, engineers need to calculate angular velocities and accelerations, which are most conveniently expressed using radians. Similarly, in physics, the phase of a wave is often expressed in radians, making it easier to analyze wave interference and diffraction patterns.
