A Journal for Western Man-- Issue XII
G. Stolyarov II
It is a common property of mathematics that, for a given angle theta, the tangent of theta/2 is equal to radical([1-cosine theta]/[1+ cosine theta]) This property has been of assistance in determining numerical relationships between the tangents of two angles whose measurements differ by a factor of two, essentially opening the way for a systematic exploration and derivation of tangent values by means of system, instead of guesswork. Yet this property bears with it a definite inconvenience, that is, the need for knowledge of cosine theta prior to its usage. Within a limited scope, that is, in relation to angles of Fundamental Pythagorean Triples (FPTs), this will no longer be a necessity.
Recall our algorithms for general derivation of FPTs from Stolyarov�s Eighth Corollary:
a=2R+d
b= (2R^2)/d+2R
c= (2R^2)/d+2R+d
Now, let us define our terms. Let alpha denote the angle opposite side a, and let beta denote the angle opposite side b. Hence, by the identity of the tangent, tan alpha= a/b = [2R+d]/[(2R^2)/d+2R]= (2Rd+d^2)/(2R^2+2Rd)= [d(2R+d)]/[R(2R+2d)]. The tangent of beta equals the inverse, b/a= [R(2R+2d)]/ [d(2R+d)]. Cosine alpha will equal b/c= [(2R^2)/d+2R]/[ (2R^2)/d+2R+d], and cosine beta will equal a/c= (2R+d)/ [(2R^2)/d+2R+d].
Presently, is would be possible to substitute values specific to FPTs into the general formula for tan(alpha/2).
Tan (alpha/2)= radical({1-[(2R^2)/d+2R]/[ (2R^2)/d+2R+d]}/{1+[(2R^2)/d+2R]/[ (2R^2)/d+2R+d]})= radical ([(2R^2)/d+2R+d-(2R^2)/d-2R]/[(2R^2)/d+2R+d+(2R^2)/d+2R])= radical(d/[4R^2/d+4R+d])= radical (d^2/(2R+d)^2) = d/(2R+d)
No cosines necessary here! A mere knowledge of radial numbers and macroperiodic difference numbers, which can be soundly derived via methods shown in previous corollaries for any given FPT, will suffice to establish a proportion between tan alpha and tan (alpha/2).
The ratio tan alpha/ tan (alpha/2) is [d(2R+d)]/[R(2R+2d)]/ [d/(2R+d)]= [d(2R+d)^2]/[Rd(2R+2d)]= (2R+d)^2/[2R(R+d)]
Knowing R and d values, it will be a matter of seconds to determine a scalar multiple which in the above form will transform tan (alpha/2) into tan alpha and, in inverse form, determine tan (alpha/2) from tan alpha .
Tan (beta/2)= radical ({1-(2R+d)/ [(2R^2)/d+2R+d]}/{1+(2R+d)/ [(2R^2)/d+2R+d]})= (after the elimination of the cumbersome denominators within the fraction) radical ({2R^2/d}/{2R^2/d+4R+2d})= radical (2R^2/[2(R+d)^2])= R/(R+d)
The ratio tan beta/ tan (beta/2) is [R(2R+2d)]/ [d(2R+d)]/ [R/(R+d)] = 2(R+d)^2/[d(2R+d)]
The applications for this corollary can alleviate an immense burden of calculations. Pretend that you are dividing an FPT triangle into two by means of a segment stretching from the vertex of the right angle to the c side. It may or may not be a median of an altitude, but the segment nevertheless does produce two triangles with inscribed circles tangent to either a or b. Now, the radius of the circle tangent to either a or b forms a kite with that tangent to c, whose other two sides are a segment of either a or b and a segment of c. This kite can be split into two triangles each containing either angle alpha/2 or beta/2, whose tangent ratio can elementarily be derived via the method which considers radial periodicity. After the tangent ratio is known, a simple �walk-around� approach can be used to determine the side of the triangle which is not the radius, and the radius of the circle can be obtained by use of the tangent ratio. This approach would, henceforth, conserve several steps of the process for time better spent.
(C) 2003 G. Stolyarov II, All rights reserved.
Stolyarov's Theorem and all related concepts are property of G. Stolyarov II.
