I have explained in another comment why this is so.
This misconception about angles being dimensionless quantities is unfortunately taught in many school textbooks and it cripples the thinking about physics of many people.
The foundation of physics is the theory of the measurement of physical quantities. However, when teaching physics frequently this is taught either badly or not at all, instead of following classical examples, like that of James Clerk Maxwell, who started his Treatise on Electricity and Magnetism with an exposition of the theory of the measurement of physical quantities that was complete and up-to-date for that time.
The plane angle a.k.a. phase angle not only is not a dimensionless quantity, but its unit of measurement plays a crucial role in determining the other base units of any modern system of units of measurement.
The reason why the unit of plane angle is so important is that plane angle is the only fundamental continuous quantity for which it is possible to approximate measurement by counting, because the numeric value of a plane angle measured in cycles can be decomposed in a sum of a discrete quantity, the integer part of the numeric value, with a continuous quantity, the fractional part of the numeric value. The value of the discrete quantity that is the integer part of the numeric value of an angle measured in cycles can be obtained by counting.
This unique property is the cause that for the highest possible precision in measurements all continuous quantities are converted by various methods into phase angles, before the analog-to-digital conversion.
The fact that plane angle has a natural unit, which is the cycle (or its integer multiples or submultiples) is exploited in defining almost all units of other continuous quantities.
For instance, the unit of length is said to be the wavelength of a certain wave. Because wavelength is a physical quantity equal to the ratio between length and plane angle, the previous definition stated correctly says that the unit of length is equal to the unit of plane angle multiplied with a certain wavelength (i.e. 1 meter = 1 cycle multiplied by a wavelength measured in meter per cycle). Similarly for many other continuous quantities, whose units are also derived in one form or another from the unit of plane angle, because only it can be defined intrinsically, instead of being derived from other units.
I have explained in another comment why this is so.
This misconception about angles being dimensionless quantities is unfortunately taught in many school textbooks and it cripples the thinking about physics of many people.
The foundation of physics is the theory of the measurement of physical quantities. However, when teaching physics frequently this is taught either badly or not at all, instead of following classical examples, like that of James Clerk Maxwell, who started his Treatise on Electricity and Magnetism with an exposition of the theory of the measurement of physical quantities that was complete and up-to-date for that time.
The plane angle a.k.a. phase angle not only is not a dimensionless quantity, but its unit of measurement plays a crucial role in determining the other base units of any modern system of units of measurement.
The reason why the unit of plane angle is so important is that plane angle is the only fundamental continuous quantity for which it is possible to approximate measurement by counting, because the numeric value of a plane angle measured in cycles can be decomposed in a sum of a discrete quantity, the integer part of the numeric value, with a continuous quantity, the fractional part of the numeric value. The value of the discrete quantity that is the integer part of the numeric value of an angle measured in cycles can be obtained by counting.
This unique property is the cause that for the highest possible precision in measurements all continuous quantities are converted by various methods into phase angles, before the analog-to-digital conversion.
The fact that plane angle has a natural unit, which is the cycle (or its integer multiples or submultiples) is exploited in defining almost all units of other continuous quantities.
For instance, the unit of length is said to be the wavelength of a certain wave. Because wavelength is a physical quantity equal to the ratio between length and plane angle, the previous definition stated correctly says that the unit of length is equal to the unit of plane angle multiplied with a certain wavelength (i.e. 1 meter = 1 cycle multiplied by a wavelength measured in meter per cycle). Similarly for many other continuous quantities, whose units are also derived in one form or another from the unit of plane angle, because only it can be defined intrinsically, instead of being derived from other units.