Step 1: Built a suitable diagram for the given question  

Speed of the rain v_r = 35 m/s, direction N to S

Speed of the bicycle v_b = 12 m/s , Direction E to S

Direction of the umbrella

Step 1: Obtain the Direction of the resultant vector (umbrella)

Relative velocity of the rain with respect to cyclist

v_{rb} = v_r - v_b

Direction of the resultant vector:

The formula for direction can be calculated using the inverse tangent on the component ratios: or \tan \theta = . components ratio

From the figure: Angle between the vertical and resultant = \theta

Therefore, \tan \theta = \frac{opp}{adj} = \frac{v_b}{v_r}

\tan \theta = \frac{12}{35}   

\theta = \tan^{-1} (\frac{12}{35})

\theta = 19\degree

Hence, the woman can keep her umbrella at an angle of about 19° with the vertical towards the west.