Other than looking at a unit circle, what is the best method to solve for the bounds of a limaçon? For example given the problem:
Find the area of the inner loop of $$\frac{1}{2} + \cos(\theta)$$
I know that to find the first value it would be $\arccos(-\frac{1}{2})$, but how would I get the second bound?
Hint:
The limacon is a polar curve of the form $$r = b + a\cos \theta$$ For $b < a$ (this is the case here) $$A = \frac{1}{2} \int_{\pi-\theta_0}^{\pi+\theta_0} (b+a\cos \theta)^2 d\theta = \int_{\pi-\theta_0}^{\pi} (b+a\cos \theta)^2 d\theta = (\frac{1}{2}a^2+b^2)\cos^{-1}(\frac{b}{a}) -\frac{3}{2}b\sqrt{a^2-b^2}$$