Barik transform and Barik eliminant: a novel algebraic method to solve polynomial equations in the sine and cosine of one variable

Systems of trigonometric equations appear frequently in problems related to geometry, kinematics, computer graphics, robotics, and astronomy, among many other fields. Most often these are first converted to algebraic equations via the ``tangent half-angle'' substitution and solved subsequently using various algebraic methods, such as computation of resultants. In the process, parametric singularities are introduced in the equations, and their degrees are also doubled. In this article, a novel concept call the ``Barik transform'' is introduced, which can help solve such a system without encumbering these issues, albeit only in systems that contain {\em at least} one equation of the first degree in the sine and cosine of the variable to be eliminated. It is shown how Barik transform can produce extremely compact eliminants in the closed form, without resorting to any resultant computation.