A Journal for Western Man




Stolyarov's Twelfth Corollary:

Derivations of Angular Tangent

Relationships Using Radial Periodicity of

Fundamental Pythagorean Triples

G. Stolyarov II

Issue XII- April 9, 2003


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: 

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.

G. Stolyarov II is a science fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, contributor to Enter Stage Right, Le Quebecois Libre, Rebirth of Reason, and the Ludwig von Mises Institute, Senior Writer for The Liberal Institute, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress. His newest science fiction novel is Eden against the Colossus. His latest non-fiction treatise is A Rational Cosmology. Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.

This TRA feature has been edited in accordance with TRA’s Statement of Policy.

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Learn about Mr. Stolyarov's novel, Eden against the Colossus, here.

Read Mr. Stolyarov's new comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, at http://www.geocities.com/rational_argumentator/rc.html.