006 Components of a force in axes that are not perpendicular to each other

From the figure:
$\phi = 180^\circ - 157^\circ = 23^\circ$

$\varphi = 215^\circ - 180^\circ = 35^\circ$
 

 

$\theta = 215^\circ - 157^\circ = 58^\circ$

$\alpha = 69^\circ - \phi = 69^\circ - 23^\circ = 46^\circ$

$\beta = 180^\circ - 69^\circ - \varphi = 180^\circ - 69^\circ - 35^\circ = 76^\circ$
 

Check:
$\theta + \alpha + \beta = 58^\circ + 46^\circ + 76^\circ = 180^\circ$       (okay!)
 

By Sine Law:
$\dfrac{P_u}{\sin \beta} = \dfrac{P_v}{\sin \theta} = \dfrac{P}{\sin \alpha}$

$\dfrac{P_u}{\sin 76^\circ} = \dfrac{P_v}{\sin 58^\circ} = \dfrac{50}{\sin 46^\circ}$

$P_u = \dfrac{50 \, \sin 76^\circ}{\sin 46^\circ} = 67.44 \, \text{ kN}$           answer

$P_v = \dfrac{50 \, \sin 58^\circ}{\sin 46^\circ} = 58.95 \, \text{ kN}$           answer