It is not possible to work without at least one dimensional measurement. The 1900 m on the mountain is not enough unless it somehow is relation to the elevation of the cathedral and the bridge.
I think it can be done the following way:
draw a horizontal with the cathedral on the W side and the bridge on the E side.
draw a vertical to represent the mountain's height of 1900 m between these two.
Then the angle of elevation from the bridge to the mountain top is 42º and the angle of elevation from the cathedral is 38º.
so by trig we can find the two sides of our two right angled triangle.
I got 2110.16 m and 2431.89 m for a total distance of 4542 m
my bum bum
1 - find the dist. from the vertical base of the mountain to the cathedral and bridge via trig ratios (tan). remember that the angle of depression = looking horizontally ahead and taking the angle between your gaze and the target.
2 - use pythagoras to find the straight-line dist.
Note that you could not use this method if the cathedral and the bridge were not east-north of the base of the mountain.
The question could be formulated differently to make it more understandable. Instead of mountain, they could have put tower (or anything else that is completely vertical)
I would think that if you draw a line to represent the middle of the mounain and then draw a line 42 degrees to the right, then draw a nother line 38 degrees to the right. Because the height of the mountain is 1900 m, then you can just divide it by 100 to get 19 cm and then when you find where the line meets, times the amount back by 100 to get what measurement in m, the straight line is. I hope that helps. :D