Checking units for equation with degree symbol

Question

Using the following equation:

$$ U = \left(\frac{B \times L \times \sin(\theta)}{C}\right)^{1/3} $$

I can calcukate the velocity of a flow traveling down a slope.

I would like to check that the units I'm using in this equation are correct. Therefore I do some checks:

$$ ms^{-1} = \left(\frac{m^{2} s^{-3} \times m \times ^{o}}{-}\right)^{1/3} $$

note $C$ is dimensionless. If I ignore the degree symbol for $\sin(\theta)$ this equation makes sense i.e. the units on the left and right match up. How does this work without ignoring the degree unit? Would you even consider the degree when doing this? Note that $\theta$ is the angle of a slope.

Answer 1

The angle $\theta$ might have units of degrees, yes. It could also have been in e.g. radians. But putting it through the sine function removes the unit. The $\sin(\theta)$ is unitless.

The sine and cosine functions are defined as the "distance" vertically and horizontally, respectively, to the point on the unit circle. That is, a distance per unit lenght. That is unitless. By using these functions you get a distance without units, not an angle.

Answer 2

According to the SI Brochure as well as ISO 80000, plane angle is an ISQ derived quantity.

The SI coherent derived unit of the plane angle expressed in terms of SI base units is ${\text{m/m}}$. Thus, expressed in terms of other SI units, the SI coherent derived unit is simply $1$.

The radian is a special name for the number one that may be used to convey information about the quantity concerned. The special symbol is ${\text{rad}}$.

$${\text{rad}} = {\text{m/m}} = 1$$