Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$.

Question

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$. Restrict the values of $p$ in $[0,2\pi]$.

The given equation has real roots if: $$\cos^2 p \geq 4\sin p (\cos p -1)$$ and we now need to find the range of values of $p$. I have tried manipulating the inequality but have not been able to find the range of $p$. Any suggestions? Also are their any other methods other than using the inequality?

Thanks...

Answer 1

As I mention in the comments, the values of $p$ for which a real root(s) exist(s) isn't very "clean", but there is a cyclical range within which it holds. A good way to approximate these intervals is to graph the right-hand side and the left-hand side to obtain the points at which the equality holds, to determine the intervals' endpoints:

These points of intersection are given by:

Lower bound:

Upperbound