The question reads Calculate the Length of PS
First notice that $\displaystyle P = 90$ degrees
Use the Sin Rule
$\displaystyle \frac{Sin P}{p} = \frac{Sin Q}{q}$
$\displaystyle \frac{Sin 90}{10} = \frac{Sin 55}{q}$
$\displaystyle q = 8,1915 = PR$
$\displaystyle Sin 35 = \frac{PS}{PR} = \frac{PS}{8,1915}$
$\displaystyle PS = (8,1915) \times sin 35 = 4,6985$ approx.
We don't even need trig. (That does not imply the thread is posted here.)
Some special relations in this case: $\displaystyle \overline{PQ}^{2}=\overline{QS}\cdot \overline{QR},\,\overline{PR}^{2}=\overline{RS}\cd ot \overline{RQ},\,\overline{PS}^{2}=\overline{SQ}\cd ot \overline{SR}.$