Question

chapter 5 review
score: 40/100 answered: 4/10
question 3
the angle between 0 and 2π in radians that is coterminal with the angle $\frac{39}{8}pi$ in radians is
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Explanation:

Step1: Recall coterminal - angle formula

Coterminal angles are given by $\theta\pm 2k\pi$, where $k$ is an integer. We want to find $k$ such that $0\leq\theta + 2k\pi\leq2\pi$. Let $\theta=\frac{39}{8}\pi$. Then we set up the inequality $0\leq\frac{39}{8}\pi+ 2k\pi\leq2\pi$.

Step2: Solve the left - hand side of the inequality for $k$

First, solve $0\leq\frac{39}{8}\pi+2k\pi$.
\[

$$\begin{align*} 0&\leq\frac{39}{8}\pi + 2k\pi\\ - \frac{39}{8}\pi&\leq2k\pi\\ -\frac{39}{16}&\leq k \end{align*}$$

\]

Step3: Solve the right - hand side of the inequality for $k$

Next, solve $\frac{39}{8}\pi+2k\pi\leq2\pi$.
\[

$$\begin{align*} \frac{39}{8}\pi+2k\pi&\leq2\pi\\ 2k\pi&\leq2\pi-\frac{39}{8}\pi\\ 2k\pi&\leq\frac{16 - 39}{8}\pi\\ 2k\pi&\leq-\frac{23}{8}\pi\\ k&\leq-\frac{23}{16} \end{align*}$$

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

Step4: Calculate the coterminal angle

Substitute $k=-2$ into $\theta + 2k\pi$.
\[

$$\begin{align*} \frac{39}{8}\pi+2\times(- 2)\pi&=\frac{39}{8}\pi-4\pi\\ &=\frac{39}{8}\pi-\frac{32}{8}\pi\\ &=\frac{39 - 32}{8}\pi\\ &=\frac{7}{8}\pi \end{align*}$$

\]

Answer:

$\frac{7}{8}\pi$