How do you evaluate arctan((arctan(9/7) - arctan(7/6)) / (arctan(5/3) - arctan(3/2)))?

1 Answer
Apr 1, 2016

pi/4+K*pi, K in NN

Explanation:

Let's rewrite the expression as

epsilon=arctan(theta/rho)
with
theta=arctan(9/7)-arctan(7/6) => tan theta=tan(arctan(9/7)-arctan(7/6))
rho=arctan(5/3)-arctan(3/2) => tan rho=tan (arctan(5/3)-arctan(3/2))

Using the formula of the tangent of the difference between two angles
tan(x-y)=(tanx-tany)/(1+tanx*tany)
we get

tan theta=(9/7-7/6)/(1+9/cancel(7)*cancel(7)/6)=((54-49)/42)/(1+3/2)=2/cancel5*cancel5/42=1/21
-> theta=arctan(1/21)
and
tan rho=(5/3-3/2)/(1+5/cancel3*cancel3/2)=((10-9)/6)/(1+5/2)=2/7*1/6=1/21
-> rho=arctan(1/21)

So
epsilon=arctan(theta/rho)=arctan(cancel(arctan(1/21))/cancel(arctan(1/21)))=arctan(1)
=> epsilon=pi/4+K*pi, K in NN