How does the value of an inverse cosine function change when the unit of its argument changes?

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.
If you’re interested, think this problem through on your own. Then check Problem of the Day 86 at my other site, on or after 23 April 2013, for a discussion.


About Santo D'Agostino

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. Further resources, and online tutoring, can be found at my other site
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2 Responses to How does the value of an inverse cosine function change when the unit of its argument changes?

  1. Mario Le Blanc says:

    Wouldn’t the short answer be: ‘it is improper to plug values with units into mathematical functions, only dimensionless values should be used.’? For instance, the propagation factor for a radio wave is omega*t – beta*r, respectively frequency*time and wave number*position, both dimensionless quantities.

    • Yes, agreed, with some clarification needed. For example, it’s OK to calculate the area of a rectangle using the formula A = xy, where each of x and y have units attached, and A “inherits” the appropriate units used for x and y. So the question is, in which mathematical functions are we allowed to use units, and in which not? Certainly the arguments of inverse trigonometric, logarithmic, and exponential functions must be unitless, to list some of the commonly used functions.

      I didn’t notice this issue being discussed in the textbooks I consulted, so I thought it would be worthwhile to point this out explicitly to students. The two quantities in the question are meaningless, but I hope that reflecting on them will lead students to the correct conclusion; if not, a discussion will appear on my other site tomorrow.

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