In figure,

V_m is speed of man=8 km/hr

V_M is increased speed of man=12 km/hr

V_r= speed of rain

V_R= speed of rain,with increased speed of man

We have,

V_rain-road=V_rain-man+V_man-road ......(1)

Let, velocity of rain with respect to road

V= u i+v j

from the first condition,

V_rain-road=V_rain-man+V_man-road

So,u i+v j=V_rain-man+8 i .......(2)

Now put V_rain-man= V¹

u=8 km/hr (because Vrain-man is vertically downward)

Now from second condition,

u i+v j=V¹ cos30 i-V¹ sin30 i+12 i  .....(3)

So,we have

u=12-V¹ sin30=8    .......(4)

V¹ sin30=4

V¹=8

V=V¹cos30  ......(5)

V=V¹cos30 =8×√3/2 =4√3

So,V_rain-road=8 i+4√3j

and tan α=4√3/8 =√3/2