Answer
$$
\begin{array}{|c|c|c|}
\hline
\theta\ (\text{degrees}) & \sin(\theta) & \tan(\theta) \\
\hline
35^\circ & 0.574 & 0.700 \\
25^\circ & 0.423 & 0.466 \\
18^\circ & 0.309 & 0.325 \\
8^\circ & 0.139 & 0.141\\
\hline
\end{array}
$$
a. $\theta = 25^\circ$
b. $\theta = 8^\circ$
Work Step by Step
Use a calculator to find the value of $\sin\theta, \tan\theta$, the difference between them, and the relative difference:
$$
\begin{array}{|c|c|c|c|c|}
\hline
\theta\ (\text{degrees}) & \sin(\theta) & \tan(\theta) & \tan(\theta) - \sin(\theta) & \frac{\tan(\theta) - \sin(\theta)}{\sin(\theta)} \times 100\% \\
\hline
35^\circ & 0.574 & 0.700 & 0.127 & 22.08\% \\
25^\circ & 0.423 & 0.466 & 0.044 & 10.34\% \\
18^\circ & 0.309 & 0.325 & 0.016 & 5.15\% \\
8^\circ & 0.139 & 0.141 & 0.001 & 0.98\%\\
\hline
\end{array}
$$
a. When does the difference become approximately $10\%$ of $\sin\theta$?
From the table: At $25^\circ$, the relative difference is approximately $10.34\%$.
b. When does the difference become approximately $1\%$ of $\sin\theta$?
From the table: At $8^\circ$, the relative difference is approximately $1\%$.
