How does the value of $\cos^{-1} \left ( 0.1 \, \textrm{cm} \right )$ differ from $\cos^{-1} \left ( 0.1 \, \textrm{m} \right )$?

This is an interesting question, especially because the question of units in trigonometric and inverse trigonometric functions is rarely discussed in mathematics textbooks. It is discussed in physics textbooks, so you may or may not have thought about this in the past. If you have never thought about it, it would be worthwhile to go away and think about this for some time, and then come back to continue reading the discussion here.

OK, now that you have thought about it, think about a trigonometric function, such as the cosine function. The cosine of some angle is the ratio of two sides of a triangle, which are always measured in the same unit, so that the ratio of the two sides is unitless; the units cancel upon division. Thus, the value of a cosine function is unitless. It’s the same for the other trigonometric functions.

Because the value of a trigonometric function is unitless, the only arguments (the quantities in the parentheses) for inverse trigonometric functions that make sense are also unitless. Thus, the two quantities listed in the first line of this post **do not make sense**.

It’s worth thinking about the other functions commonly used in mathematics; which of them have arguments that can have units, and which of them have arguments that must be unitless?

(A version of this post first appeared at my other (now deleted) blog, and was transferred to this blog on 20 January 2021.)