The sine function can be represented as \(\sin (\theta)=\theta-\frac{\theta^{3}}{3 !}+\frac{\theta^{5}}{5 !}-\frac{\theta^{7}}{7 !}+\frac{\theta^{9}}{9 !} \cdots\). Plot the percent error between \(\sin (\theta)\) and:

- \(\theta(\mathrm{rad})\)

- \(\theta-\frac{\theta^{3}}{3 !}(\mathrm{rad})\)

- \(\theta-\frac{\theta^{3}}{3 !}+\frac{\theta^{5}}{5 !}(\mathrm{rad})\)

for a range of \(\theta\) values from \(0.001 \mathrm{rad}\) to \(\frac{\pi}{2} \mathrm{rad}\) in steps of \(0.001 \mathrm{rad}\). Calculate the percent error using \(\left(\frac{\theta-\sin (\theta)}{\sin (\theta)}\right) \cdot 100\) for the \(\theta\) approximation, \(\left(\frac{\left(\theta-\frac{\theta^{3}}{3}\right)-\sin (\theta)}{\sin (\theta)}\right) \cdot 100\) for the \(\theta-\frac{\theta^{3}}{3 !}\) approximation, and so on. How do these results relate to the small angle approximation?