# Give the smallest value for 'a' to complete the graph

The curve $$r=5\sinθ$$ is given for $$0\leθ\le a$$
It appears that if $$a = \pi$$, the graph is complete. Now my question is why? Since I thought it should be $$a = 2\pi$$ since $$2\pi = 360°$$, which is a full circle. ## 1 Answer

If you plot the points at some spacing of $$\theta$$ you will see the graph is complete when $$\theta=\pi$$. Because of the relationship $$\sin (\theta+\pi)=-\sin (\theta)$$ as $$\theta$$ rises from $$\pi$$ to $$2\pi$$ you have $$r$$ being negative, which reflects the angle through the center and retraces the graph you have already plotted.

• But if the sin is negative, then theta is also negative right? Which is not allowed. – Stallmp Jul 11 at 14:55
• Usually $\theta$ is allowed to run over $(-\infty, \infty)$ and you plot enough of it to get what you want. That doesn't matter in this case. $\sin \frac {3\pi}2$ is negative even though $\frac {3\pi}2$ is not – Ross Millikan Jul 11 at 15:05
• I see, so sin can be negative while theta remains strictly positive? – Stallmp Jul 11 at 15:37
• Sure. Just plot $y=\sin x$ and you will see. It is negative on $(\pi, 2\pi)$ and on all intervals $2k\pi$ above and below – Ross Millikan Jul 11 at 17:52