Aguy wire makes a 67° angle with the ground. walking out 32 feet further from the tower, the angle of elevation to the top of the tower is 39°. find the height of the tower.

This is the theory of geometry, the height of the tower will be given by:
tan theta=[opposite]/[adjacent]
let the opposite be h;
adjacent be x;
tan 67=h/x
thus;
h=xtan67
also
tan39=h/(x+32)
h=(x+32)tan39
xtan67=(x+32)tan39
2.4x=(x+32)0.8
2.4x=0.8x+25.6
2.4x-0.8x=25.6
1.6x=25.6
x=25.6/1.6
x=16 ft
thus
h=xtan67
h=16tan67
h=37.7 ft

Geometric pattern of 2 i’m guessing. not really sure about what you’re asking but i hope this !