Question

an arc on a circle measures $85^\circ$. 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 $\pi$ radians

Explanation:

Step1: Convert degrees to radians

The formula to convert degrees to radians is $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$. For $85^\circ$:
$\text{Radians} = 85 \times \frac{\pi}{180} = \frac{17\pi}{36} \approx 1.484$

Step2: Define range values

Calculate the numerical values of the range bounds:

• $\frac{\pi}{2} \approx 1.571$
• $\pi \approx 3.142$
• $\frac{3\pi}{2} \approx 4.712$

Step3: Compare to find range

$\frac{17\pi}{36} \approx 1.484$, which is greater than $0$ and less than $\frac{\pi}{2} \approx 1.571$.

Answer:

0 to $\frac{\pi}{2}$ radians