Answer
$6.67\%; 4.03\%$
Work Step by Step
The percentage error for the first case is
$percentage \space error=\frac{\theta_{r, Ptolemy}-\theta_r}{\theta_r}\times \% 100$
$percentage \space error=\frac{8.00^{\circ}-7.50^{\circ}}{7.50^{\circ}}\times \%100=6.67\%$
We know that
$sin\theta_r=\frac{(1.00)sin 20^{\circ}}{1.33}$
$sin\theta_r=0.2572$
$\implies \theta_r=14.90^{\circ}$
Now, the percentage error can be determined as
$percentage \space error=\frac{\theta_{Ptolemy}-\theta_r}{\theta_r}\times 100\%$
$\implies percentage \space error =\frac{15.5^{\circ}-14.90^{\circ}}{14.90^{\circ}}\times 100\%$
$\implies percentage \space error=4.03\%$
