To find an angle between \(0\) radians and \(2\pi\) radians that is coterminal with \(-\frac{4\pi}{5}\) radians, you can add \(2\pi\) to \(-\frac{4\pi}{5}\) until the result falls within the desired range.
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Calculate \(2\pi\) in terms of a fraction with the same denominator as \(-\frac{4\pi}{5}\): \[ 2\pi = \frac{10\pi}{5} \]
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Add \(2\pi\) to \(-\frac{4\pi}{5}\): \[ -\frac{4\pi}{5} + \frac{10\pi}{5} = \frac{6\pi}{5} \]
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Now, check if \(\frac{6\pi}{5}\) is within the range \(0\) to \(2\pi\): \[ 0 < \frac{6\pi}{5} < 2\pi \] Since \(2\pi = \frac{10\pi}{5}\), we see that \( \frac{6\pi}{5} < \frac{10\pi}{5} \).
Thus, the angle that is coterminal to \(-\frac{4\pi}{5}\) radians and lies between \(0\) radians and \(2\pi\) radians is
\[ \frac{6\pi}{5} \text{ radians}. \]