Question

find an angle between 0 and 2\pi that is coterminal with the given angle. 9 rad

Explanation:

Step1: Recall coterminal - angle formula

Coterminal angles are of the form $\theta\pm 2k\pi$, where $\theta$ is the given angle and $k\in\mathbb{Z}$. We want to find $k$ such that $0\leq\theta + 2k\pi<2\pi$. Here $\theta = 9$.

Step2: Determine the value of $k$

We know that $2\pi\approx 6.28$. Let's find $k$ by solving the inequality $0\leq9 + 2k\pi<2\pi$.
First, solve $0\leq9 + 2k\pi$:
\[2k\pi\geq - 9\Rightarrow k\geq-\frac{9}{2\pi}\approx-\frac{9}{6.28}\approx - 1.43\]
Then, solve $9 + 2k\pi<2\pi$:
\[2k\pi<2\pi - 9\Rightarrow k<\frac{2\pi - 9}{2\pi}=1-\frac{9}{2\pi}\approx1 - 1.43=-0.43\]
Since $k\in\mathbb{Z}$, $k=-1$.

Step3: Calculate the coterminal - angle

Substitute $k = - 1$ into the coterminal - angle formula $\theta+2k\pi$.
\[9+2\times(- 1)\pi=9 - 2\pi\]

Answer:

$9 - 2\pi$