Applied Physics of Extended (or Gamma) Sine and Cosine Functions

Abstract

Evolution of fundamental mathematical tools (such as trigonometric functions sin(α) and cos(α)) has inherent repercusions
 on how we solve problems in applied physics. Recently published extended or gamma sine function sin∗(α, γ)
 and cosine function cos∗(α, γ) — along with their upgraded identity angle sum and subtraction rules sin∗(A ± B, γ) and
 cos∗(A ± B, γ) — have enabled a new approach on how to tackle practical problems using mathematics (a published example
 is the energy-coupled mass-spring oscilatory system). The usefullness of a theory is measured by both the insight
 it generates, and the solution it produces, when applied to physical problems with pertinent applications. Its acceptance
 amongst peers depends on the availability of such examples, as way-showers of how the theory can be applied in practice,
 and how useful results can be derived by employing it in similar or related examples/problems. This article has the
 purpose of providing this bridge between the above theories and its application in some common scientific fields. Several
 exercises are solved employing these new formulae, and new potential applications are identified that cover various topics
 in physics such as civil engineering (i.e., measuring distances in bridges), aerospace and aeronautics (i.e., turbine velocity
 triangles and optimum orbital deployment for a satellite constellation) and telecommunications (i.e., antenna array
 beamforming and steering, as well as new modulations based on quadrature phase-shift keying). These problems (and
 solutions) are designed to indicate the usefullness of these new expanded functions, and can become practical classroom
 exercises applicable to both academic and professional environments.