To find the exact length of the arc created by a central angle [tex]\(\theta = 60^\circ\)[/tex] in a circle with a radius of [tex]\(r = 132\)[/tex] cm, we can follow these steps:
1. Convert the central angle from degrees to radians:
A central angle in degrees can be converted to radians by using the formula:
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\][/tex]
For [tex]\(\theta = 60^\circ\)[/tex]:
[tex]\[
\theta_{\text{radians}} = 60 \times \frac{\pi}{180} = \frac{\pi}{3}
\][/tex]
2. Calculate the arc length:
The formula for the arc length [tex]\(s\)[/tex] of a circle is given by:
[tex]\[
s = r \times \theta_{\text{radians}}
\][/tex]
Substituting the known values [tex]\(r = 132\)[/tex] cm and [tex]\(\theta_{\text{radians}} = \frac{\pi}{3}\)[/tex]:
[tex]\[
s = 132 \times \frac{\pi}{3}
\][/tex]
3. Simplify the expression:
Simplify the multiplication:
[tex]\[
s = \frac{132 \pi}{3} = 44 \pi
\][/tex]
Therefore, the exact length of the arc is:
[tex]\[
s = 44 \pi \text{ cm}
\][/tex]
