a an angle measuring α radians is co - terminal with an angle measuring $ \frac{9 \pi}{2} $ radians. if $ - 2 \pi \leq \alpha \leq 0 $, what is the value of $ \alpha $?
* radians
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b. if $ 0 \leq \alpha \leq 2 \pi $, what is the value of $ \alpha $?
* radians
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c an angle measuring $ \beta $ radians is co - terminal with an angle measuring $ - 8 $ radians. if $ - 2 \pi \leq \beta \leq 0 $, what is the value of $ \beta $?
* radians
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d if $ 0 \leq \beta \leq 2 \pi $, what is the value of $ \beta $?

Question

Accepted Answer

### Part a
# Explanation:
## Step1: Recall co - terminal angles formula
Co - terminal angles differ by an integer multiple of $2\pi$. We want to find an angle $\alpha$ such that $- 2\pi\leq\alpha\leq0$ and $\alpha$ is co - terminal with $\frac{9\pi}{2}$. Let $\alpha=\frac{9\pi}{2}+k\cdot2\pi$, where $k$ is an integer. We need to find $k$ such that $-2\pi\leq\frac{9\pi}{2}+k\cdot2\pi\leq0$.

First, solve $\frac{9\pi}{2}+k\cdot2\pi\geq - 2\pi$:
$$
\begin{align*}
\frac{9\pi}{2}+k\cdot2\pi&\geq - 2\pi\
k\cdot2\pi&\geq - 2\pi-\frac{9\pi}{2}\
k\cdot2\pi&\geq-\frac{4\pi + 9\pi}{2}\
k\cdot2\pi&\geq-\frac{13\pi}{2}\
k&\geq-\frac{13}{4}=- 3.25
\end{align*}
$$

Then, solve $\frac{9\pi}{2}+k\cdot2\pi\leq0$:
$$
\begin{align*}
\frac{9\pi}{2}+k\cdot2\pi&\leq0\
k\cdot2\pi&\leq - \frac{9\pi}{2}\
k&\leq-\frac{9}{4}=-2.25
\end{align*}
$$

Since $k$ is an integer, $k = - 3$.

## Step2: Calculate $\alpha$
Substitute $k=-3$ into $\alpha=\frac{9\pi}{2}+k\cdot2\pi$:
$$
\begin{align*}
\alpha&=\frac{9\pi}{2}+(-3)\cdot2\pi\
&=\frac{9\pi}{2}-6\pi\
&=\frac{9\pi - 12\pi}{2}\
&=-\frac{3\pi}{2}
\end{align*}
$$

# Answer:
$-\frac{3\pi}{2}$

### Part b
# Explanation:
## Step1: Recall co - terminal angles formula
We want to find an angle $\alpha$ such that $0\leq\alpha\leq2\pi$ and $\alpha$ is co - terminal with $\frac{9\pi}{2}$. Let $\alpha=\frac{9\pi}{2}+k\cdot2\pi$, where $k$ is an integer. We need to find $k$ such that $0\leq\frac{9\pi}{2}+k\cdot2\pi\leq2\pi$.

First, solve $\frac{9\pi}{2}+k\cdot2\pi\geq0$:
$$
\begin{align*}
k\cdot2\pi&\geq-\frac{9\pi}{2}\
k&\geq-\frac{9}{4}=-2.25
\end{align*}
$$

Then, solve $\frac{9\pi}{2}+k\cdot2\pi\leq2\pi$:
$$
\begin{align*}
\frac{9\pi}{2}+k\cdot2\pi&\leq2\pi\
k\cdot2\pi&\leq2\pi-\frac{9\pi}{2}\
k\cdot2\pi&\leq\frac{4\pi - 9\pi}{2}\
k\cdot2\pi&\leq-\frac{5\pi}{2}\
k&\leq-\frac{5}{4}=-1.25
\end{align*}
$$

Since $k$ is an integer, $k=-2$.

## Step2: Calculate $\alpha$
Substitute $k = - 2$ into $\alpha=\frac{9\pi}{2}+k\cdot2\pi$:
$$
\begin{align*}
\alpha&=\frac{9\pi}{2}+(-2)\cdot2\pi\
&=\frac{9\pi}{2}-4\pi\
&=\frac{9\pi - 8\pi}{2}\
&=\frac{\pi}{2}
\end{align*}
$$

# Answer:
$\frac{\pi}{2}$

### Part c
# Explanation:
## Step1: Recall co - terminal angles formula
We want to find an angle $\beta$ such that $-2\pi\leq\beta\leq0$ and $\beta$ is co - terminal with $- 8$ radians. Let $\beta=-8 + k\cdot2\pi$, where $k$ is an integer. We need to find $k$ such that $-2\pi\leq-8 + k\cdot2\pi\leq0$.

First, solve $-8 + k\cdot2\pi\geq - 2\pi$:
$$
\begin{align*}
k\cdot2\pi&\geq - 2\pi + 8\
k&\geq\frac{-2\pi + 8}{2\pi}\approx\frac{-6.28+8}{6.28}\approx\frac{1.72}{6.28}\approx0.27
\end{align*}
$$

Then, solve $-8 + k\cdot2\pi\leq0$:
$$
\begin{align*}
k\cdot2\pi&\leq8\
k&\leq\frac{8}{2\pi}\approx\frac{8}{6.28}\approx1.27
\end{align*}
$$

Since $k$ is an integer, $k = 1$.

## Step2: Calculate $\beta$
Substitute $k = 1$ into $\beta=-8 + k\cdot2\pi$:
$$
\begin{align*}
\beta&=-8+2\pi
\end{align*}
$$

# Answer:
$-8 + 2\pi$ (or approximately $-8 + 6.28=-1.72$)

### Part d
# Explanation:
## Step1: Recall co - terminal angles formula
We want to find an angle $\beta$ such that $0\leq\beta\leq2\pi$ and $\beta$ is co - terminal with $- 8$ radians. Let $\beta=-8 + k\cdot2\pi$, where $k$ is an integer. We need to find $k$ such that $0\leq-8 + k\cdot2\pi\leq2\pi$.

First, solve $-8 + k\cdot2\pi\geq0$:
$$
\begin{align*}
k\cdot2\pi&\geq8\
k&\geq\frac{8}{2\pi}\approx1.27
\end{align*}
$$

Then, solve $-8 + k\cdot2\pi\leq2\pi$:
$$
\begin{align*}
k\cdot2\pi&\leq2\pi + 8\
k&\leq\frac{2\pi + 8}{2\pi}\approx\frac{6.28 + 8}{6.28}\approx\frac{14.28}{6.28}\approx2.27
\end{align*}
$$

Since $k$ is an integer, $k = 2$.

## Step2: Calculate $\beta$
Substitute $k = 2$ into $\beta=-8 + k\cdot2\pi$:
$$
\begin{align*}
\beta&=-8 + 4\pi
\end{align*}
$$

# Answer:
$-8 + 4\pi$ (or approximately $-8+12.56 = 4.56$)

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QUESTION IMAGE

Question

Explanation:

Part a

Step1: Recall co - terminal angles formula

Step2: Calculate \(\alpha\)

Step1: Recall co - terminal angles formula

Step2: Calculate \(\alpha\)

Step1: Recall co - terminal angles formula

Step2: Calculate \(\beta\)

Answer:

Part b