To solve the equation:
[tex]\[ -4 \sin x = -\cos^2 x + 4 \][/tex]
we start by rewriting it for clarity:
[tex]\[ -4 \sin x + \cos^2 x - 4 = 0 \][/tex]
We need to find the solutions to this equation within the interval [tex]\([0, 2\pi)\)[/tex]. This is a transcendental equation and solving it involves finding when the left side equals zero. The solutions can generally be complex, but we are looking for those within the given interval in terms of [tex]\(\pi\)[/tex].
After solving the equation, the solutions found are:
[tex]\[ x = -\frac{\pi}{2}, \quad x = -2 \arctan\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3}\right), \quad x = -2 \arctan\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3}\right) \][/tex]
However, we need to consider these solutions within the interval [tex]\([0, 2\pi)\)[/tex]. For real solutions in this interval, we have:
[tex]\[ x = -\frac{\pi}{2} \][/tex]
Converting to the interval [tex]\([0, 2\pi)\)[/tex], we add [tex]\(2\pi\)[/tex]:
[tex]\[ x = -\frac{\pi}{2} + 2\pi = \frac{3\pi}{2} \][/tex]
Therefore, the solution in radians in terms of [tex]\(\pi\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] is:
[tex]\[ x = \frac{3}{2}\pi \][/tex]
So the solutions to the equation in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[ x = \frac{3}{2}\pi \][/tex]
