By speaking of angles and lengths, the question is asking about the dimensions of parameters we use in doing transformations such as rotations, boosts, and translations.

The sin(x) answer recognized that angle x must be expressed in the very special mathematical quantity of radians in order for the expansion of sin(x) or cos(x), found in trigonometry or rotation matrices, to make sense. The underlying reason for this is that rotations form a non-abelian Lie Group. If you make a small rotation of something (eg: a stick) about the x axis by $\theta_x$ and then about the y axis by $\theta_y$, the final stick will not point in the same direction as one first rotated about y and then x. The two final sticks are related by a rotation about the z axis by $\theta_z = \theta_x \theta_y$ where the $\theta$ must be expressed in radians for this product to be mathematically correct. Rotations do not commute. Non-commutivity makes rotation angles, measured in radians special and dimensionless. If we insist on using a dimensioned quantity such as d[degrees] to measure rotation angles, then we need a new fundamental constant $\kappa = \frac{180}{\pi}$ [degrees] to convert d angles to radians.

$$ \theta = (\frac{d}{\kappa}) = [radians] $$

Similarly, non-commutivity of velocity boosts (Special Relativity and the Lorentz Group) causes boosts to be done in dimensionless radians. The Lorentz Boost parameter $\lambda$ [radians] replaces velocities $v$ [m/sec].

$$\lambda = tanh^{-1}(\frac{v}{c}) = [radians] $$

Like $\kappa$ for dimensioned rotation angles, the purpose of the fundamental velocity c is to turn a dimensioned boost velocity into dimensionless radians. Because c is the same everywhere, the meter or second could be removed from the Bureau of Standards, since the other can be obtained using the fundamental constant c. Let’s say the standard meter is still left in the Bureau.

This still leaves length being a dimensioned quantity, which means it is measured as multiples of the standard meter (an arbitrary length stick) stored in the Bureau of Standards (or more modernly as so many wavelengths of light from a particular atomic transition). If translations are one day found not to commute, then they too will have to be done by radians, and there will be some new fundamental constant length L to convert meters to radians. There will then be no dimensioned quantities left, therefore no dimensional analysis, and therefore no need to ask the very perceptive question that started this long winded answer.

By speaking of angles and lengths, the question is asking about the dimensions of parameters we use in doing transformations such as rotations, boosts, and translations.

The sin(x) answer recognized that angle x must be expressed in the very special mathematical quantity of radians in order for the expansion of sin(x) or cos(x), found in trigonometry or rotation matrices, to make sense. The underlying reason for this is that rotations form a non-abelian Lie Group. If you make a small rotation of something (eg: a stick) about the x axis by $\theta_x$ and then about the y axis by $\theta_y$, the final stick will not point in the same direction as one first rotated about y and then x. The two final sticks are related by a rotation about the z axis by $\theta_z = \theta_x \theta_y$ where the $\theta$ must be expressed in radians for this product to be mathematically correct. Rotations do not commute. Non-commutativity makes rotation angles, measured in radians special and dimensionless. If we insist on using a dimensioned quantity such as d [degrees] to measure rotation angles, then we need a new fundamental constant $\kappa = \frac{180}{\pi}$ [degrees] to convert d angles to radians.

$$ \theta = \left(\frac{d}{\kappa}\right) \ \text{[radians]} $$

Similarly, non-commutativity of velocity boosts (Special Relativity and the Lorentz Group) causes boosts to be done in dimensionless radians. The Lorentz Boost parameter $\lambda$ [radians] replaces velocities $v$ [m/sec].

$$\lambda = \tanh^{-1} \left(\frac{v}{c}\right) \ \text{[radians]} $$

Like $\kappa$ for dimensioned rotation angles, the purpose of the fundamental velocity $c$ is to turn a dimensioned boost velocity into dimensionless radians. Because $c$ is the same everywhere, the meter or second could be removed from the Bureau of Standards, since the other can be obtained using the fundamental constant $c$. Let’s say the standard meter is still left in the Bureau.

This still leaves length being a dimensioned quantity, which means it is measured as multiples of the standard meter (an arbitrary length stick) stored in the Bureau of Standards (or more modernly as so many wavelengths of light from a particular atomic transition). If translations are one day found not to commute, then they too will have to be done by radians, and there will be some new fundamental constant length L to convert meters to radians. There will then be no dimensioned quantities left, therefore no dimensional analysis, and therefore no need to ask the very perceptive question that started this long winded answer.