Answer
$\theta$ lies in the third quadrant.
Work Step by Step
$sec~\theta\lt0$:
$\frac{1}{cos~\theta}\lt0$
$cos~\theta\lt0$
$tan~\theta\gt0$
$\frac{sin~\theta}{cos~\theta}\gt0$
Then $sin~\theta\lt0$ because both $sin~\theta$ and $cos~\theta$ must have the same sign.
We can conclude that $\theta$ lies in the third quadrant.
