section_path: Measuring angles/Dimensional analysis
section_name: Dimensional analysis
paragraph_text: Dimensional analysis
  
  Plane angle may be defined as θ = s / r, where θ is the (numerical value of the) subtended angle in radians, s is arc length, and r is radius. One SI radian corresponds to the (numerical value of the) angle expressed in radians for which s = r, hence 1 SI radian = 1 m/m = 1. However, rad is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using the area of a circular sector θ = 2 A / r gives 1 SI radian as 1 m /m = 1. The key fact is that the SI radian is a dimensionless unit equal to 1. In SI 2019, the SI radian is defined accordingly as 1 rad = 1. It is a long-established practice in mathematics and across all areas of science to make use of rad = 1.
  
  Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimeters, where r is the numerical value of the radius of the pulley when expressed in centimeters and θ is the numerical value of the angle through which the pulley turns when expressed in radians. When multiplying r by θ the unit radian does not appear in the result. Similarly in the formula for the angular velocity of a rolling wheel, ω = v / r, radians appear in the units of ω but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".
  
  In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s), and torsional stiffness (N⋅m/rad), and not in the quantities of torque (N⋅m) and angular momentum (kg⋅m /s).
  
  At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the area of a circle, π r. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.
  
  In particular, Quincey identifies Torrens' proposal to introduce a constant η equal to 1 inverse radian (1 rad) in a fashion similar to the introduction of the constant ε. With this change the formula for the angle subtended at the center of a circle, s = rθ, is modified to become s = ηrθ, and the Taylor series for the sine of an angle θ becomes: Sin θ = sin x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = η θ − (η θ) 3 3 ! + (η θ) 5 5 ! − (η θ) 7 7 ! + ⋯, {\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta)^{3}}{3!}}+{\frac {(\eta \theta)^{5}}{5!}}-{\frac {(\eta \theta)^{7}}{7!}}+\cdots,} where x = η θ = θ / rad {\displaystyle x=\eta \theta =\theta /{\text{rad}}} is the numerical value of the angle when expressed in radians.The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while sin is the traditional function on pure numbers which assumes its argument is a dimensionless number in radians. The capitalised symbol Sin {\displaystyle \operatorname {Sin} } can be denoted sin {\displaystyle \sin } if it is clear that the complete form is meant.
  
  Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1. This radian convention allows the omission of η in mathematical formulas.