Answer
$\sec 314^{\circ} 40'$ = 1.4225
Work Step by Step
Given function-
$\sec 314^{\circ} 40'$
We have to convert it to decimal degrees first-
$\sec 314^{\circ} 20'$= $\sec (314^{\circ} + 40')$
= $\sec (314^{\circ} + (\frac{40}{60})^{\circ})$
= $\sec (314^{\circ} + (0.6667)^{\circ})$
= $\sec 314.6667^{\circ} $
Using calculator in degree mode-
314.6667 → $\cos$ → $\frac{1}{x}$ (Recall $\sec\theta$ = $\frac{1}{\cos\theta}$)
$\sec 314.6667^{\circ}$ = 1.4225125995
Rounding it to four places past the decimal point-
$\sec 314.6667^{\circ}$ = 1.4225
or
$\sec 314^{\circ} 40'$ = 1.4225
