Question

find a positive angle less than ( 2pi ) that is coterminal with the given angle.
( -\frac{21pi}{5} )
a positive angle less than ( 2pi ) that is coterminal with ( -\frac{21pi}{5} ) is
(simplify your answer. type your answer in terms of ( pi ). use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall coterminal angle formula

To find a coterminal angle with a given angle \(\theta\), we add or subtract multiples of \(2\pi\) (since the period of the trigonometric functions is \(2\pi\)). For a negative angle, we add multiples of \(2\pi\) until we get a positive angle less than \(2\pi\). Let's denote the given angle as \(\theta = -\frac{21\pi}{5}\). We need to find an integer \(n\) such that \(\theta + n\cdot 2\pi> 0\) and \(\theta + n\cdot 2\pi< 2\pi\).

First, let's find how many times we need to add \(2\pi\) (which is \(\frac{10\pi}{5}\) to have a common denominator with \(-\frac{21\pi}{5}\)). Let's solve for \(n\) in the inequality:

\(-\frac{21\pi}{5}+n\cdot 2\pi>0\)

\(n\cdot 2\pi>\frac{21\pi}{5}\)

Divide both sides by \(\pi\) (since \(\pi> 0\), the inequality sign remains the same):

\(2n>\frac{21}{5}\)

\(n > \frac{21}{10}=2.1\)

Since \(n\) must be an integer, we start with \(n = 3\) (because \(n = 2\) gives \(2\times2 = 4\), \(\frac{21}{5}=4.2\), \(4<4.2\), so \(n = 2\) is not enough). Let's check \(n = 3\):

Step2: Calculate the coterminal angle

\(\theta_{coterminal}=-\frac{21\pi}{5}+3\times2\pi\)

First, calculate \(3\times2\pi=\frac{30\pi}{5}\)

Then, \(\theta_{coterminal}=-\frac{21\pi}{5}+\frac{30\pi}{5}=\frac{- 21\pi + 30\pi}{5}=\frac{9\pi}{5}\)

Now, check if \(\frac{9\pi}{5}<2\pi\) (since \(2\pi=\frac{10\pi}{5}\), and \(\frac{9\pi}{5}<\frac{10\pi}{5}\), so it satisfies the condition of being less than \(2\pi\)).

Answer:

\(\frac{9\pi}{5}\)