$$\\sec(\delta) = -6/5 tan(\delta)\\\\\\
\frac{1}{cos(\delta)}}=\frac{-6Sin(\delta)}{5cos(\delta)}\\\\
so\qquad \delta \ne n\pi\;\;radians \quad n\in Z\qquad and\\\\
1=\frac{-6Sin(\delta)}{5}\\\\
\frac{5}{-6}=sin(\delta)\\\\
\delta $must be in 3rd or 4th quadrant$\\
$but tan has to be neg so $\delta $ must be in the 4th quad $\\\\
\delta=asin(\frac{-5}{6})$$
$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{6}}}}\right)} = -{\mathtt{56.442\: \!690\: \!238\: \!079^{\circ}}}$$
$$\delta \approx -56^0\pm 360N\qquad where \quad N\in Z$$
.
$$\\sec(\delta) = -6/5 tan(\delta)\\\\\\
\frac{1}{cos(\delta)}}=\frac{-6Sin(\delta)}{5cos(\delta)}\\\\
so\qquad \delta \ne n\pi\;\;radians \quad n\in Z\qquad and\\\\
1=\frac{-6Sin(\delta)}{5}\\\\
\frac{5}{-6}=sin(\delta)\\\\
\delta $must be in 3rd or 4th quadrant$\\
$but tan has to be neg so $\delta $ must be in the 4th quad $\\\\
\delta=asin(\frac{-5}{6})$$
$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{6}}}}\right)} = -{\mathtt{56.442\: \!690\: \!238\: \!079^{\circ}}}$$
$$\delta \approx -56^0\pm 360N\qquad where \quad N\in Z$$