The orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of 1 z. For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1 x and the side opposite has an orientation 1 y. Since (using ~ to indicate orientational equivalence) tan(θ) = θ + ... ~ 1 y /1 x we conclude that an angle in the xy-plane must have an orientation 1 y /1 x = 1 z, which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation 1 z while cos(θ) has orientation 1 0. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos(θ) + b sin(θ), where a and b are real scalars. An expression such as sin ⁡ ⁡ ( θ θ + π π / 2 ) = cos ⁡ ⁡ ( θ θ ) is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written: