Mickey determines that the angle of elevation from his position to the top of the tower is 52°. He measures the angle of elevation again from a point 47 meters farther from the tower and finds to be 31°. Both positions are due East of the tower. Find the height of the tower.

Answer 1

Your previous 3 posts are all single trig ratios in a right-angled triangle. You MUST know the basic 3 trig ratios and how they are applied in such cases.
This one is the only one that requires more than one step.

make a sketch, mark the top of the tower P and its base Q
Mark his first position A, and his 2nd position B
Fill in the information,

In triangle ABP,
angle A = 31°
angle B = 180-52 = 128°
then angle P = 180-128-31 = 21°

by the Sine Law:
BP/sin31 = 47/sin21
BP = 47sin31/sin21 = .....

Now back to the right-angled triangle PBQ
PQ/PB = sin52
PQ = BPsin52
= (you know BP from above)
or in one string of calculations
= (47sin31)(sin52)/sin21 = appr 53.23 m
or 53 m to the nearest metre