On Fundamental Errors in Trigonometry

Abstract

The critical analysis of the foundations of standard trigonometry is proposed. The unity of formal logic and  rational dialectics is methodological basis of the analysis. The analysis leads to the following main results: (1)  trigonometry does not treat a right triangle as a material system. Therefore, trigonometry does not satisfy the  system principle; (2) trigonometric functions do not satisfy the mathematical definition of a function. The terms  “sine”, “cosine”, “tangent”, “cotangent” and others are not identical to the concept of function. Symbols “cos”,  “sin”, “tg”, “ctg”, etc. indicate only that there is a correspondence (connection) between the values of the  quantities of the angle and the lengths of the sides in a right-angled triangle. Therefore, the standard definitions  of trigonometric functions do not represent mathematical (quantitative) relationships between the quantities of  the angle and the lengths of the sides in a right-angled triangle. Trigonometric functions are neither explicit nor  implicit functions; (3) the range of definition of trigonometric functions does not satisfy the condition for the  existence of a right-angled triangle because the definitions of trigonometric functions contradict to the system  principle. These facts prove the assertion that the trigonometric functions, the trigonometric identities, the  trigonometric form of the Pythagorean Theorem and the inverse trigonometric functions are blunders; (4) the  values of mathematical quantities are always neutral numbers. Therefore, logical contradictions arise if the  quantity of the angle and the symbols “cos”, “sin”, “tg”, “ctg” take on negative values. (5) it is proved that the  standard theorems of addition (difference) of two arguments for cosine and sine are blunders. This means that  the addition (difference) theorems for all trigonometric functions, the reduction formula, the formula for double  and half argument are blunders; (6) in the point of view of the Cartesian coordinate system, the abscissa and  ordinate scales are identical and have the dimension “meter”. Therefore, the quantity of the angle (which has the  dimension “degree”) does not exist in the Cartesian coordinate system; (7) the graphs of trigonometric functions  are built in an inadmissible coordinate system because the scales are not identical: the abscissa scale has the  dimension “degree”, and the ordinate scale has the dimension “meter”. The non-identity of the dimensions leads  to absurdity: “meter” is “degree”. Therefore, the graphs of trigonometric functions have no geometric meaning;  (8) if the material point is the end point of the moving radius in the material system “circle + mobile radius +  Cartesian coordinate system”, then the graph of the dependence of the ordinate of the material point on the  length of the path traveled (i.e., on the circumference of a given radius) has the form of a sinusoid, but the graph  is not a trigonometric sinusoid. Consequently, standard trigonometry is a pseudoscientific theory.