The equation in fig.1 can be transformed to the one in fig.2. Using ‘atan’, we can turn the periodic function ‘tan’,which is on the left of Eq.1, to the sum of ‘atan’ and l*π. However, for the case of fig.3, such method cannot be applied.
Anyway to combine the two ‘cot’s on the right? If not, any methods to eliminate the periodic functions in Eq.3, and gain a function only contain aperiodic functions and the term l*π. The first two equations derive from the PDF below(Eq 70/71).Much obliged!
Hi, for Eq. 3 M=a cotan b + b cotan a, you may try to use the Taylor expansion of the cotan function, of the form [1/x + infinite sum sigma 2x/(x^2 - n^2.pi^2)] (n varies from 0 to infinity); then simplify and write the term under the summation sign sigma in the form of sum of [C/(b - n.pi) + D/(a-n.pi)], basically the "decomposition in partial fraction" where n appears with power 1 in denominators instead of power 2 and where you find C and D; after the decomposition in fraction, use the "telescoping series" evaluation technique by searching the limit of the sigma sum. This should evaluate the sum. Let me know if any roadblock, and at which level as I have not got time to do the entirely math myself, I can only foresee the solution using that 3 steps strategy of (1) infinite sum from Taylor expansion, (2) decompose in fraction and (3) evaluation as for telescoping series. You may also evaluate the infinite sum other wise. Hope this helps
If the three quantities a, b and M are such that the two ratios a/M and b/M belong to the interval [-1, 1] then these two ratios can be denoted as cos(c) and cos(d), respectively, i.e., cos(c) = a/M and cos(d) = b/M.
Thus, the right-hand-side (RHS) in eq. 3 can be written as sumation of two products involving two cosines and a sine:
RHS(numerator) = cos(c) cos(b) sin(a) + cos(d) cos(a) sin(b)
with denominator sin(a) sin(b) that may go in the LHS.
Then, one identity can be applied for modifying the RHS products into a sum:
for every three real x, y, z it is true that
4 sin(x) cos(y) cos(z) =
= sin(x+y+z) + sin(x+y-z) + sin(x-y+z) + sin(x-y-z),
however, the trigonometric sums should be further worked out ...
A lot of algebraic manipulations are needed here, thus such idea may or may not lead to anything valuable.
Well if your purpose is to remove all the periodic (trigonometric) functions, then it can be done in simple ways.
1. Convert your expression into Sine and Cosine
2. Use the Sine rule to convert Sine into sides of the triangle and Circumradius.
3. Use the Cosine rule to convert Cosines into Sides of the triangle.
In this way, you can get rid of the periodic functions and find your expression in terms of the sides of the triangle.
I hope it will work for you.
Thank you
Indeed, a strategy using the law of tangent (hence cotan by taking inverses), and maybe the Mollweide’s formulae. This may add additional conditions and how they fit in a triangle.
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