# Question #23b7a

Jun 21, 2017

$\theta = \frac{\pi}{30} + \frac{2 \pi}{5} \cdot n , \frac{11 \pi}{30} + \frac{2 \pi}{5} \cdot n$, $n = 0 , 1 , 2 , 3 , \ldots$

#### Explanation:

$\cos 5 \theta = \frac{\sqrt{3}}{2}$

$5 \theta = {\cos}^{- 1} \left(\frac{\sqrt{3}}{2}\right)$

$5 \theta = \frac{1}{6} \pi , \frac{11}{6} \pi , \frac{13}{6} \pi , \frac{23}{6} \pi , \frac{25}{6} \pi , \frac{35}{6} \pi \ldots$

$\theta = \frac{1}{30} \pi , \frac{11}{30} \pi , \frac{13}{30} \pi , \frac{23}{30} \pi , \frac{25}{30} \pi , \frac{35}{30} \pi \ldots$

we can divide into 2 type of sequences.
$\theta = \frac{1}{30} \pi , \frac{13}{30} \pi , \frac{25}{30} \pi , \ldots$ $\to i$
$\theta = \frac{1}{30} \pi , \frac{1}{30} \pi + \frac{12}{30} \pi , \frac{1}{30} \pi + \frac{24}{30} \pi , \ldots$
$\theta = \frac{1}{30} \pi + \frac{12}{30} n \cdot \pi$

$\theta = \frac{1}{30} \pi + \frac{2}{5} n \cdot \pi$, $n = 0 , 1 , 2 , 3 , \ldots$

$\theta = \frac{\pi}{30} + \frac{2 \pi}{5} \cdot n$, $n = 0 , 1 , 2 , 3 , \ldots$

$\theta = \frac{11}{30} \pi , \frac{23}{30} \pi , \frac{35}{30} \pi , \ldots$ $\to i i$
$\theta = \frac{11}{30} \pi , \frac{11}{30} \pi + \frac{12}{30} \pi , \frac{11}{30} \pi + \frac{24}{30} \pi , \ldots$
$\theta = \frac{11}{30} \pi + \frac{12}{30} n \cdot \pi$,

$\theta = \frac{11}{30} \pi + \frac{2}{5} n \cdot \pi$, $n = 0 , 1 , 2 , 3 , \ldots$

$\theta = \frac{11 \pi}{30} + \frac{2 \pi}{5} \cdot n$, $n = 0 , 1 , 2 , 3 , \ldots$

Jun 21, 2017

$t = \pm \frac{\pi}{30} + \frac{2 k \pi}{5}$

#### Explanation:

$\cos 5 t = \frac{\sqrt{3}}{2}$
Use trig table and unit circle:
$5 t = \pm \frac{\pi}{6} + 2 k \pi$
$t = \pm \frac{\pi}{30} + \frac{2 k \pi}{5}$
$\left(- \frac{\pi}{30}\right)$ by the co-terminal arc $\frac{59 \pi}{30}$.
$t = \frac{\pi}{30} + \frac{2 k \pi}{5}$
$t = \frac{59 \pi}{30} + \frac{2 k \pi}{5}$