Opposition and adjacent means we are using Tangent
Let h be the height and x be the distance along the ground to the first measurement.
tan 52.8 = h / x
so h = x (tan 52.8)
And tan 34.9 = h / (x + 8)
so h = (x + 8)(tan 34.9)
This means that x (tan 52.8) = (x + 8)(tan 34.9). We need to isolate x.
Distribute: x (tan 52.8) = x (tan 34.9) + 8(tan 34.9)
Subtract: x (tan 52.8) - x (tan 34.9) = 8(tan 34.9)
Factor: x (tan 52.8 - tan 34.9) = 8(tan 34.9)
Divide: x = 8(tan 34.9) / (tan 52.8 - tan 34.9)
Calculate: x = 9.00371 = 9 meters
Use x to get h:
h = 9.00371(tan 52.8) = 11.8619
And the height of the survey instrument: 11.86 + 1.7 = 13.56 = 13.6 meters.
