[USMA:22364] The Deceptive and Unnecessary Radian

[david clayton]

The application of the radian unit has introduced ambiguity and contradictions in logic to mathematics.   The removal of the radian by the formalised use of circular measure (using an abbreviation "circ" in equations) takes away the source of these complications.   Examples of ambiguity: Example 1. "If an angle x is small and measured in radians, sin x approximately equals x." Comment: The x in sin x is an angle and can be expressed in any angle units and is not restricted to radians. The simple x value is not an angle, it is the purely numerical circular measure of the angle, and can be evaluated from an angle specified in any units. The corrected statement is : "If an angle x is small, sin x approximately equals circ x." Also, angles are converted to radians rather than "measured in radians", because of the impractical size of the radian.   Example 2. The current use of  "angle measured in radians" actually must be interpreted as "angle or its circular measure as appropriate". Comment: For instance, in Calculus when the derivations are examined, it is seen that expressions are integrated and differentiated with respect to the circular measure of an angle, and "angle measured in radians" is substituted for it.   Examples of Contradiction: Example 1. The radian is given the dimensionless value 1.0. Comment: This dimensionless value contradicts the ability of polar co-ordinates to describe two-dimensional and three-dimensional space, using the dimensional value of angle units and a single length dimension.   Example 2. When functions such as sin theta(Greek letter) are plotted, the label used for the x-axis can vary disconcertingly when the radian is involved. Three labelling methods have been noticed: Method (a) The x-axis is labelled theta/rad. Method (b) The x-axis is labelled theta. Method (c) The x-axis is labelled theta(rad). Comment: The x-axis shows values of circ theta   Replacing the Radian as a base unit of the International System Desirable features of a base unit include: 1. Convenience in use. 2. If not always of a convenient size, it should be easily divided or multiplied into units of convenient size. 3. Should already be widely known. 4. Should constantly have angular dimensions. (The radian had none of these features)   An angle of one revolution seems a candidate.