Physical Interpretation of Complex Angles and Their Functions

A child's idea of an angle is a corner or edge; the sharper the corner or edge, the smaller is the angle, and hence expressions like “acute (sharp) angle,” “obtu e (blunt) angle,” etc. With a little schooling his idea of an angle is broadened into that of turning or rotation and finds a geometrical expression in a circle. The amount of turning is represented by the area of the sector of the circle like a piece of pie gone over by the turning radius. This broadened dea is still in agreement with the earlier idea because the sector is a surface having a corner at the center of the circle, and the sharpness or bluntness of this corner varies with the area. With further progress, he also learns to associate the angle with the arc of the circle, for, evidently, the length of the arc is a measure of the amount of turning and is proportional to the area of the sector. The angle when represented by the arc of the circle loses all resemblance to a corner, which latter idea however is discarded as unnecessary, emphasis now being laid on “turning” which is well represented by the arc of the circle. Mathematical analysis is then applied which still further broadens the scope of angles and gives rise to the so called “imaginary” angles which are called “hyperbolic” in contrast with ordinary “real” angles which are called “circular,” combinations of these two kinds of angles being called “general” or “complex.” At this stage, any physical interpretation or geometrical representation of an angle practically vanishes, for, a hyperbolic angle can not be conceived of as a rotation, not even as a rotation around an “imaginary” axis, for, how can one conceive of an “imaginary” axis? If the “imaginary” axis be defined as a new axis at right angles to the old axis, rotation around it can easily be conceived of, but it does not correspond to facts, because a hyperbolic angle implies no change in direction, and hence no rotation, but merely affects the magnitude of a quantity. Inasmuch as problems can be solved mathematically without any visualization, the mathematician discards visual interpretations and bases his conceptions on formulas. However, the engineering type of mind finds it both difficult and distasteful to be dependent on symbolic definitions, and craves for visualization. He feels that if physical problems lead to “imaginary” or “complex” angles, these angles must stand for some concrete physical facts and must therefore be capable of a physical interpretation. A broad and yet simple physical interpretation applicable to both circular and hyperbolic angles is developed below, which also applies to their trigonometric functions, as follows: A quantity may be expressed or specified either by its own dimensions or as a percentage of another quantity. The first kind of specification is its “absolute” measure, the second kind is its “angular” measure. Thus “angle” and “percentage” are synonymous terms. If a quantity changes by a certain percentage (compound percentage as in compound interest) that “percentage” is the “angular” change in the quantity in radians. According to this interpretation, neither corner, edge, turning nor rotation are essential characteristics of an angle; they are the characteristics of a limited class only. If the percentage change in a quantity is like itself, geometrically parallel to itself, the per cent change, that is, the angle, is called a hyperbolic angle, and there is no rotation. If the quantity varies without any change in magnitude but only with a change in direction, geometrically perpendicular to itself, the per cent change, that is, the angle, is called a circular angle, and there is rotation. Thus, hyperbolic and circular angles are at right angles to each other, the first being a per cent change parallel to the quantity, and the second perpendicular to it. “Imaginary” thus means a quadrature. However, it is not rotation around a quadrature axis but per cent change in a quadrature direction. This new interpretation is in agreement with the representation by arcs in a more extended vectorial sense in which one is not limited to the arc of a circle but arcs of any shape have an angular interpretation. Thus, consider a portion of an arc of any arbitrary shape, and resol e it into two components, one circular (at right angles to the radius drawn to the point) and the other straight and radial (parallel to the radius at that point). These two arcs expressed as a percentage of the radius represent circular and hyperbolic angles, respectively, and being in quadrature with each other their vector sum is the “complex” angle of the arc. In this interpretation a hyperbola has nothing to do with a hyperbolic angle, and the name is unfortunate. It would have been more intelligible to speak of the ordinary circular angle as a “quadrature” angle, and of the so called hyperbolic angle as a “linear” angle or a “radial” angle. This new interpretation of an angle is then utilized to interpret the physical significance of sines and cosines of angles, whether circular or hyperbolic. The common idea of a cosine as the projection of the radius vector on the X axis, and of sine as the projection on the Y axis, is found inadequate, this idea being limited to circular angles and not being of much value in explaining their occurrence in physical problems. A cosine is then interpreted as the mean vector value of a unit quantity for a given per cent increase and an equal per cent decrease, or a given value of rotation positive and negative. Examples are given illustrating the application of this vector idea in physical problems, as in the resolution of a harmonic wave into two oppositely rotating vectors, solution of differential equations, etc. A geometrical construction is given for the sines and cosines of complex angles, which is not merely a representation but photographically true, so to speak.