Question

find the angle between 0 and 2π in radians that is coterminal with the angle 14/3 π. enter your answer as a reduced fraction involving π.

Explanation:

Step1: Use the coterminal angle formula

Coterminal angles can be found using the formula \(\theta + 2k\pi\), where \(k\) is an integer. We want to find \(k\) such that \(0\leqslant\frac{14}{3}\pi+ 2k\pi<2\pi\).
Let's solve the inequality \(0\leqslant\frac{14}{3}\pi+2k\pi\) and \(\frac{14}{3}\pi + 2k\pi<2\pi\).
First, consider \(\frac{14}{3}\pi+2k\pi\geqslant0\), which is always true for negative \(k\) (since \(\frac{14}{3}\pi>0\) and \(2k\pi\) is negative for \(k < 0\)).
Now consider \(\frac{14}{3}\pi+2k\pi<2\pi\).
Subtract \(\frac{14}{3}\pi\) from both sides: \(2k\pi<2\pi-\frac{14}{3}\pi=\frac{6\pi - 14\pi}{3}=-\frac{8\pi}{3}\).
Divide both sides by \(2\pi\): \(k<-\frac{4}{3}\).
Since \(k\) is an integer, let \(k=- 2\).

Step2: Calculate the coterminal angle

Substitute \(k =-2\) into \(\theta=\frac{14}{3}\pi+2k\pi\).
\(\theta=\frac{14}{3}\pi+2\times(-2)\pi=\frac{14}{3}\pi - 4\pi=\frac{14\pi-12\pi}{3}=\frac{2\pi}{3}\).

Answer:

\(\frac{2\pi}{3}\)