Question

an arc on a circle measures 295°. the measure of the central angle, in radians, is within which range?
○ 0 to $\frac{\pi}{2}$ radians
○ $\frac{\pi}{2}$ to $\pi$ radians
○ $\pi$ to $\frac{3\pi}{2}$ radians
○ $\frac{3\pi}{2}$ to $2\pi$ radians

Explanation:

Step1: Recall the conversion formula

To convert degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Step2: Convert \( 295^\circ \) to radians

Substitute \( 295^\circ \) into the formula: \( 295 \times \frac{\pi}{180} = \frac{59\pi}{36} \approx 5.15 \) radians.

Step3: Analyze the ranges

• \( 0 \) to \( \frac{\pi}{2} \) (≈1.57) radians: \( 5.15 \) is not in this range.
• \( \frac{\pi}{2} \) (≈1.57) to \( \pi \) (≈3.14) radians: \( 5.15 \) is not in this range.
• \( \pi \) (≈3.14) to \( \frac{3\pi}{2} \) (≈4.71) radians: \( 5.15 \) is not in this range.
• \( \frac{3\pi}{2} \) (≈4.71) to \( 2\pi \) (≈6.28) radians: \( 5.15 \) is in this range.

Answer:

\( \frac{3\pi}{2} \) to \( 2\pi \) radians (the last option)