Solve sinalpha<cosvarphi

$$\sin \alpha < \cos \varphi$$

Solve for φ

$$\phi \in \left(\frac{2\arcsin(\sin(\alpha ))+4\pi n_{2}-\pi }{2},\frac{-2\arcsin(\sin(\alpha ))+4\pi n_{2}+\pi }{2}\right)$$

$$n_{2}\in \mathrm{Z}$$

$$\alpha \leq 2\pi n_{1}+\frac{\pi }{2}\text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(\alpha >2\pi n_{3}+\frac{\pi }{2}\text{ and }\alpha <2\pi n_{3}+\frac{5\pi }{2}\right)$$

Solve for α

$$\alpha \in \left(-\arcsin(\cos(\phi ))+2\pi n_{1}-\pi ,\arcsin(\cos(\phi ))+2\pi n_{1}\right)$$

$$n_{1}\in \mathrm{Z}$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\phi >2\pi n_{2}-\pi \text{ and }\phi <2\pi n_{2}+\pi \right)$$