Question

find a positive angle less than 2π that is coterminal with the given angle.
\frac{11pi}{3}
a positive angle less than 2π that is coterminal with \frac{11pi}{3} is (square). (simplify your answer. type your answer in terms of π. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall coterminal - angle formula

Coterminal angles differ by \(2k\pi\), where \(k\in\mathbb{Z}\). We want to find \(k\) such that \(0<\frac{11\pi}{3}+ 2k\pi<2\pi\).

Step2: Solve the inequality for \(k\)

Let's solve \(\frac{11\pi}{3}+2k\pi < 2\pi\). First, subtract \(\frac{11\pi}{3}\) from both sides: \(2k\pi<2\pi - \frac{11\pi}{3}=\frac{6\pi - 11\pi}{3}=-\frac{5\pi}{3}\). Then \(k<-\frac{5}{6}\). Now solve \(\frac{11\pi}{3}+2k\pi>0\). Subtract \(\frac{11\pi}{3}\) from both sides: \(2k\pi>-\frac{11\pi}{3}\), so \(k >-\frac{11}{6}\approx - 1.83\). Since \(k\in\mathbb{Z}\), \(k = - 1\).

Step3: Calculate the coterminal angle

Substitute \(k=-1\) into \(\theta=\frac{11\pi}{3}+2k\pi\). We get \(\theta=\frac{11\pi}{3}+2\times(- 1)\pi=\frac{11\pi}{3}-2\pi=\frac{11\pi - 6\pi}{3}=\frac{5\pi}{3}\).

Answer:

\(\frac{5\pi}{3}\)