Step 1: List what the inlet triangle needs.
Three things fix the triangle at inlet: the blade speed $u$, the flow speed $V_{f1}$, and the whirl speed $V_{w1}$ set by the guide vane.
Step 2: Get the blade speed from the RPM.
$u = \pi D N / 60 = \pi(1)(600)/60 = 31.42$ m/s.
Step 3: Get the whirl speed from the guide blade angle.
Since the guide angle is measured from the tangential direction, $V_{w1} = V_{f1} \cot(15^{\circ}) = 10 \times 3.732 = 37.32$ m/s.
Step 4: Build the runner triangle and solve for the vane angle.
The relative whirl left after subtracting blade speed is $37.32 - 31.42 = 5.90$ m/s, so $\tan \beta = 10/5.90 = 1.693$, giving $\beta = 59.4^{\circ}$.
Final Answer:
The runner inlet vane angle is about $59.4^{\circ}$.
\[ \boxed{\beta \approx 59.4^{\circ}} \]