i) ROD / GS = tan(theta) = 16000/(39*6080),
i.e. ROD = GS*16000/(39*6080) = 1231 fpm
calculating theta was unnecessary here
ii) 1:60 rule is an approximation, actually it ought to be 1:57.3, another approximation used is
for small theta
tan(theta) = theta (radian)
one overestimates other underestimates so we have a fairly reasonable answer.
This approximation is used in calculating theta, and calculating theta is not necessary.
iii) Unitary method is by far the most accurate method you have used... and it is same as using good trigonometry without approximation hence same answer 1231 fpm..