Question

find a positive angle less than ( 2pi ) that is coterminal with the given angle.
( \frac{35pi}{9} )
a positive angle less than ( 2pi ) that is coterminal with ( \frac{35pi}{9} ) is
(simplify your answer. type your answer in terms of ( pi ). use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles are given by \(\theta + 2k\pi\), where \(k\in\mathbb{Z}\). We want to find \(k\) such that \(0<\frac{35\pi}{9}+ 2k\pi<2\pi\).

Step2: Solve the inequality for \(k\)

First, rewrite \(2\pi=\frac{18\pi}{9}\). Then \(\frac{35\pi}{9}+2k\pi<\frac{18\pi}{9}\). Subtract \(\frac{35\pi}{9}\) from both sides: \(2k\pi<\frac{18\pi - 35\pi}{9}=-\frac{17\pi}{9}\). Divide by \(2\pi\): \(k<-\frac{17}{18}\). Also, \(\frac{35\pi}{9}+2k\pi>0\). Subtract \(\frac{35\pi}{9}\): \(2k\pi>-\frac{35\pi}{9}\). Divide by \(2\pi\): \(k>-\frac{35}{18}\approx - 1.94\). Since \(k\in\mathbb{Z}\), \(k = - 1\).

Step3: Calculate the coterminal angle

Substitute \(k=-1\) into \(\theta + 2k\pi\). We get \(\frac{35\pi}{9}+2(-1)\pi=\frac{35\pi}{9}-\frac{18\pi}{9}\).

$$ \frac{35\pi - 18\pi}{9}=\frac{17\pi}{9} $$

Answer:

\(\frac{17\pi}{9}\)