Answer
$\theta$ = $9^{\circ}9'$
Work Step by Step
Given, $\sec\theta$ = $1.0129$
We will calculate $\cos\theta$ first as calculator does not have $\sec^{-1}$ key.
Using reciprocal identity-
$\cos\theta$ = $\frac{1}{\sec\theta}$ = $\frac{1}{1.0129}$
Using calculator-(1.0129 → $\frac{1}{x}$)
$\cos\theta$ = $0.9872642907$
Therefore-
$\theta$ = $\cos^{-1} 0.9872642907$
Given $\theta$ is between $0^{\circ}$ and $90^{\circ}$, i.e. lies in QI
Using calculator in degree mode-(0.9872642907 → $\cos^{-1}$)
$\theta$ = $(9.1540061634) ^{\circ}$
$\theta$ = $9^{\circ} + (0.1540061634)^{\circ}$
$\theta$ = $9^{\circ} +(0.1540061634\times60)'$
(Recall $1^{\circ}$ = $60'$)
$\theta$ = $9^{\circ} +9'$ (Rounding to the nearest minute)
$\theta$ = $9^{\circ}9'$
