# Question #94f10

Mar 28, 2016

$\theta = \left(\frac{1}{3}\right) \left(\pm \left(2 n + 1\right) \pi - \frac{\pi}{2}\right)$, n = 0, 1, 2, 3..= $\frac{\pi}{6} , 5 \frac{\pi}{6} , 9 \frac{\pi}{6} , 13 \frac{\pi}{6} , \ldots$. The negative solutions are $- 3 \frac{\pi}{6} , - 7 \frac{\pi}{6} , - 11 \frac{\pi}{6} , \ldots .$

#### Explanation:

The general value of ${\cos}^{- 1} \left(- 1\right)$ = an odd multiple of $\pi$.

So, $3 \theta + \frac{\pi}{2}$ = an odd multiple of $\pi$ = $\pm \left(2 n + 1\right) \pi$, n = 0, 1, 2, 3, ...

Positive solutions are $\frac{\pi}{6} , 5 \frac{\pi}{6} , 9 \frac{\pi}{6} , 13 \frac{\pi}{6} , \ldots$
The negative solutions are $- 3 \frac{\pi}{6} , - 7 \frac{\pi}{6} , - 11 \frac{\pi}{6} , \ldots .$

Mar 28, 2016

$t = \frac{\pi}{2} + k . \pi$

#### Explanation:

Trig unit circle gives -->
$\cos \left(3 t + \frac{\pi}{2}\right) = - 1 = \cos \pi$
$3 t + \frac{\pi}{2} = \pm \pi$
a. $3 t = \pi - \frac{\pi}{2} = \frac{\pi}{2}$. --->$t = \frac{3 \pi}{2}$
b. $3 t = - \pi - \frac{\pi}{2} = - \frac{3 \pi}{2}$, or $\left(\frac{\pi}{2}\right)$ --> co-terminal
$t = \frac{\pi}{2} + k . \pi$