let $ heta=\frac{16pi}{7}$. complete parts (a), (b), and (c) below. (a) sketch $ heta$ in standard position. (b) find an angle between 0 and $2pi$ that is coterminal with $ heta$. write your answer in radians in terms of $pi$. radians (c) find an angle between $- 2pi$ and 0 that is coterminal with $ heta$. write your answer in radians in terms of $pi$. radians

Question

let $	heta=\frac{16pi}{7}$. complete parts (a), (b), and (c) below. (a) sketch $	heta$ in standard position. (b) find an angle between 0 and $2pi$ that is coterminal with $	heta$. write your answer in radians in terms of $pi$. radians (c) find an angle between $- 2pi$ and 0 that is coterminal with $	heta$. write your answer in radians in terms of $pi$. radians

Accepted Answer

# Explanation: ## Step1: Recall coterminal - angle formula Coterminal angles of $\theta$ are given by $\theta + 2k\pi$, where $k\in\mathbb{Z}$. Given $\theta=\frac{16\pi}{7}$. ## Step2: Find a coterminal angle between 0 and $2\pi$ for part (b) We want to find $k$ such that $0<\frac{16\pi}{7}+2k\pi < 2\pi$. First, solve the left - hand inequality $0<\frac{16\pi}{7}+2k\pi$: $$ \begin{align*} 0&<\frac{16\pi}{7}+2k\pi\ - \frac{16\pi}{7}&<2k\pi\ -\frac{8}{7}&-\frac{8}{7}\ k&>-\frac{15}{7}\approx - 2.14 \end{align*} $$ Then, solve the right - hand inequality $\frac{16\pi}{7}+2k\pi < 0$: $$ \begin{align*} \frac{16\pi}{7}+2k\pi&<0\ 2k\pi&<-\frac{16\pi}{7}\ k&<-\frac{8}{7}\approx - 1.14 \end{align*} $$ Since $k\in\mathbb{Z}$, $k=-2$. $$ \begin{align*} \theta_{2}&=\frac{16\pi}{7}+2(-2)\pi\ &=\frac{16\pi}{7}-4\pi\ &=\frac{16\pi - 28\pi}{7}\ &=-\frac{12\pi}{7} \end{align*} $$ # Answer: (b) $\frac{2\pi}{7}$ (c) $-\frac{12\pi}{7}$

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Question

Explanation:

Step1: Recall coterminal - angle formula

Step2: Find a coterminal angle between 0 and $2\pi$ for part (b)

Step3: Find a coterminal angle between $-2\pi$ and 0 for part (c)

Answer: