Question
Find angle between 0 radians and 2 pie radians that’s coterminal to -4 pie over 5 radians
Answers
GPT-4o mini
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.
1. Calculate \(2\pi\) in terms of a fraction with the same denominator as \(-\frac{4\pi}{5}\):
\[
2\pi = \frac{10\pi}{5}
\]
2. Add \(2\pi\) to \(-\frac{4\pi}{5}\):
\[
-\frac{4\pi}{5} + \frac{10\pi}{5} = \frac{6\pi}{5}
\]
3. 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
\[
\frac{6\pi}{5} \text{ radians}.
\]
1. Calculate \(2\pi\) in terms of a fraction with the same denominator as \(-\frac{4\pi}{5}\):
\[
2\pi = \frac{10\pi}{5}
\]
2. Add \(2\pi\) to \(-\frac{4\pi}{5}\):
\[
-\frac{4\pi}{5} + \frac{10\pi}{5} = \frac{6\pi}{5}
\]
3. 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
\[
\frac{6\pi}{5} \text{ radians}.
\]