See the attached image for a drawing. The points are defined as such
A = first observation point
B = second observation point (after moving 15 ft closer to the tree)
C = base of the tree
D = top of the tree
Based on the points given, we can say
segment AB = 15 ft
segment BC = x ft
segment CD = h ft
segment AC = (segment AB)+(segment BC) = (15+x) ft
where x and h are unknowns at this point. The ultimate goal is to find h, which is the height of the tree.
Focus on triangle ACD. The opposite and adjacent sides are CD and AC respectively for the reference angle 46 degrees
tan(angle) = opp/adj
tan(46) = CD/AC
tan(46) = h/(15+x)
h = (15+x)*tan(46)
Similarly, for triangle BCD, we can say
tan(angle) = opp/adj
tan(59) = CD/BC
tan(59) = h/x
h = x*tan(59)
Based on those two resulting equations, we can equate the two right hand sides to find x
(15+x)*tan(46) = x*tan(59)
15*tan(46)+x*tan(46) = x*tan(59)
15*tan(46) = x*tan(59)-x*tan(46)
15*tan(46) = x*(tan(59)-tan(46))
x*(tan(59)-tan(46)) = 15*tan(46)
x = (15*tan(46))/(tan(59)-tan(46))
x = 24.704533196337
Which is then used to find h
h = x*tan(59)
h = 24.704533196337*tan(59)
h = 41.115247719711
The approximate height of the tree is roughly 41.115247719711 feet. Round this value however you need to.
