Question

arc cd is \\(\frac{2}{3}\\) of the circumference of a circle. what is the radian measure of the central angle?\
\\(\frac{2\pi}{3}\\) radians\
\\(\frac{3\pi}{4}\\) radians\
\\(\frac{4\pi}{3}\\) radians\
\\(\frac{3\pi}{2}\\) radians

Explanation:

Step1: Recall the full circle in radians

A full circle has a central angle of \(2\pi\) radians (since the circumference corresponds to a central angle of \(2\pi\) radians).

Step2: Calculate the central angle for arc CD

Arc CD is \(\frac{2}{3}\) of the circumference. So, the central angle \(\theta\) (in radians) is \(\frac{2}{3}\) of \(2\pi\).
Mathematically, \(\theta=\frac{2}{3}\times2\pi=\frac{4\pi}{3}\) radians.

Answer:

\(\frac{4\pi}{3}\) radians (corresponding to the option " \(\boldsymbol{\frac{4\pi}{3}}\) radians")