Conference rusure::math

1647.0. "solution for general arctan equation?" by STAR::ABBASI (i^(-i) = SQRT(exp(PI))) Fri Jul 24 1992 18:12

    how to solve for w in this?
    
    arctan(a_1 w)+arctan(a_2 w)+...+arctan(a_n w) = angle.
    
    a_i, i=1..n   are known (real numbers), angle is given (in radians)
    
    I know how to do this for i up to 2 like this:
    
    if
        arctan(a_1 w)+arctan(a_2 w) = A
    then
        tan(A) = tan( arctan(a_1 w)+arctan(a_2 w) )
    
    this is like 
        tan(A) = tan(C+D)
    
    then go from here your trignometry identities.
    but how to generalize it?
    
    I must admit I only thought about this for few minutes and gave up.
    (not that i thought more I would have solved it!)
    
    I'll think more about this , but if someone knows how to solve this 
    could you please post it?
    
    thanks,
    /nasser
    

If you have:
        A = sum(k=1...n) arctan(a_k*w)

then:

	  sum(t odd) s_n,t*(iw)^t
tan(A) = --------------------------
	 i*sum(t even) s_n,t*(iw)^t

where s_n,t is the sum of the products of {a_k}, taken j at a time.
	Eg: s_3,3 = a_1*a_2*a_3
	    s_3,2 = a_1*a_2 + a_1*a_3 + a_2*a_3
	    s_3,1 = a_1 + a_2 + a_3
Note:  s_n,0 is taken to be 1, which is sensible if you think about it.

Proof by induction is easy, using the trig identity for 2 variables:
             tanx+tany
tan(x+y) = -------------
	   1 - tanx.tany

Andrew