Answer
$\theta$ = $8^{\circ}8'$
Work Step by Step
Given, $\csc\theta$ = $7.0683$
We will calculate $\sin\theta$ first as calculator does not have $\csc^{-1}$ key.
Using reciprocal identity-
$\sin\theta$ = $\frac{1}{\csc\theta}$ = $\frac{1}{7.0683}$
Using calculator-(7.0683 → $\frac{1}{x}$)
$\sin\theta$ = $0.1414767342$
Therefore-
$\theta$ = $\sin^{-1} 0.1414767342$
Given $\theta$ is between $0^{\circ}$ and $90^{\circ}$, i.e. lies in QI
Using calculator in degree mode-(0.1414767342 → $\sin^{-1}$)
$\theta$ = $(8.1333075009) ^{\circ}$
$\theta$ = $8^{\circ} + (0.1333075009)^{\circ}$
$\theta$ = $8^{\circ} +(0.1333075009\times60)'$
(Recall $1^{\circ}$ = $60'$)
$\theta$ = $8^{\circ} +8'$ (Rounding to the nearest minute)
$\theta$ = $8^{\circ}8'$
