In a study of harmonic oscillators, a phase angle $\varphi$ is found to satisfy the equation

$$4\mathrm{cosec}\varphi =9\cos \varphi$$

for $0<\varphi <\pi$. Determine the possible values of $\varphi$, in radians, giving your answers to 3 significant figures.

The configuration of a robotic linkage is governed by the equation

$$\frac{\tan 3\theta -\tan 20^{\circ }}{1+\tan 3\theta \tan 20^{\circ }}=2$$

for $0^{\circ }<\theta <180^{\circ }$. Solve this equation to find the possible orientations $\theta$, giving your answers to one decimal place.