Question

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

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 $0<\frac{16\pi}{7}+2k\pi$:

$$
LATEXBLOCK0
$$

Then, solve the right - hand inequality $\frac{16\pi}{7}+2k\pi < 2\pi$:

$$
LATEXBLOCK1
$$

Since $k\in\mathbb{Z}$, $k = - 1$.

$$
LATEXBLOCK2
$$

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

We want to find $k$ such that $-2\pi<\frac{16\pi}{7}+2k\pi < 0$.
First, solve the left - hand inequality $-2\pi<\frac{16\pi}{7}+2k\pi$:

$$
LATEXBLOCK3
$$

Then, solve the right - hand inequality $\frac{16\pi}{7}+2k\pi < 0$:

$$
LATEXBLOCK4
$$

Since $k\in\mathbb{Z}$, $k=-2$.

$$
LATEXBLOCK5
$$

Answer:

(b) $\frac{2\pi}{7}$
(c) $-\frac{12\pi}{7}$