For which values of $k$ does the trigonometric equation $\sin x + \cos^2(x) + k =0$ have no solution?
Rearranging the solutions are from the equation :
$ -\sin^2(x) + \sin(x) + 1 + k = 0$
By the fact that it is a quadratic equation in $\sin(x) $, by imposing the discriminant to be equal to zero, it should give us the range of values of k, which is:
$k < -5/4$
however, by drawing the graph of the function on Geogebra, it looks like there is another range, which I couldn't find algebraically, which is:
$k > 1 $
so that the final solutions should be:
$k < -5/4 ∨ k > 1$
THE QUESTION IS: how can i find the latter solution without looking at the graph, but simply using algebra?