Solve $\sin x - \cos x = -1$ for the interval $(0, 2\pi)$ We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.
How to solve
$$\sin x - \cos x = -1$$
for the interval $(0, 2\pi)$.

Since $\cos(\pi/4)=\sin(\pi/4)=\sqrt{2}/2$ we have using the angle sum theorem
$$\sin(x)\sin(\pi/4)-\cos(x)\cos(\pi/4)=-\sqrt{2}/2\iff\cos(x+\pi/4)=\sqrt{2}/2$$
$$\iff
x+\pi/4=\pi/4\lor x+\pi/4=7\pi/4\iff x=3\pi/2$$
as $x\in(0,2\pi)$.
