determine the value of the following if 0 is less than or equal to theta and theta is less than 2 pie.
sec theta= -2root3/3= -root/3= 7pie/6
First, it's "pi" not "pie."
Second, what exactly are you asking? These cannot all be equal to one another.
Are you asking to find $\displaystyle \theta$ when
$\displaystyle sec(\theta) = - \frac{2\sqrt{3}}{3}$
$\displaystyle sec(\theta) = -\sqrt{3}$
and to find
$\displaystyle sec \left ( \frac{7 \pi}{6} \right )$
You've been asking for help on a large number of these. Why don't you tell us what you've been able to do so we can help you learn this better.
-Dan
Okay, please in the future don't string those equal signs together like that. It is very confusing!
$\displaystyle sec(\theta) = -\frac{2\sqrt{3}}{3}$
$\displaystyle cos(\theta) = -\frac{\sqrt{3}}{2}$
Now, this implies that we have a reference angle of $\displaystyle \pi / 6$. Cosine is negative in QII and QIII. So your solution will be
$\displaystyle \theta = \pi \pm \frac{\pi}{6}$
so your final solution is both $\displaystyle 5\pi / 6$ and $\displaystyle 7 \pi / 6$.
-Dan