Units of trigonometric functions?

[John Greger]

TL;DR Summary

If I take say Sin(0.5), what would the units of the output be?

What are the units of the trigonometric functions sinus, cosinus etc? If I take say Sin(0.5), what would the units of the output be?

 

[lomidrevo]

Summary:: If I take say Sin(0.5), what would the units of the output be?

What are the units of the trigonometric functions sinus, cosinus etc? If I take say Sin(0.5), what would the units of the output be?

Why do you think they have any units?

 

[Ibix]

If I have a right angled triangle, what is $\sin$ of one angle in terms of the side lengths? What are the units of that expression?

 

[etotheipi]

In addition to @Ibix's argument, the other common argument comes from looking at the series expansions, e.g. for $\sin{x}$,$$\sin{x} = \sum_{n=0}^{\infty} u_n = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$Take any two terms $u_a$ and $u_b$ where $a \neq b$, then for dimensional homogeneity you require that $[x]^{2a+1} = [x]^{2b+1}$ but this is only satisfied when $[x] = 1$.

 

[Lnewqban]

Summary:: If I take say Sin(0.5), what would the units of the output be?

What are the units of the trigonometric functions sinus, cosinus etc? If I take say Sin(0.5), what would the units of the output be?

You could consider sines and cosines as percentages or ratios.
It is a comparison between the legths of one side and the hypotenuse.
Please, see:
https://en.m.wikipedia.org/wiki/Percentage

"A percentage is a dimensionless number (pure number); it has no unit of measurement."

 

[lomidrevo]

You could consider sines and cosines as percentages or ratios.
It is a comparison between the legths of one side and the hypotenuse.
Please, see:
https://en.m.wikipedia.org/wiki/Percentage

"A percentage is a dimensionless number (pure number); it has no unit of measurement."

Comparing trigonometric functions to percentage might be quite misleading, imho. And not applicable to tangent.

 

[PeterDonis]

not applicable to tangent.

Yes, it is: the tangent is the ratio of the lengths of the two legs (opposite to adjacent).

 

[lomidrevo]

No issue with ratios, in my post I was referring to percentage only. Let me explain more particularly what I meant ..

Tangent returns any value from $-\infty$ to $\infty$, its codomain is any real number. Although you can define a mapping between real numbers and the typical interval of percentage from 0.0% to 100.0%, I don't see any added value by doing so. I admit, "not applicable" is not the best word I could have used.
For sine and cosine, the analogy with percentage makes little bit more sense, as their return real numbers between -1.0 and 1.0, but I would need to use negative percentage, ie. interval -100.0% to 100.0%.

I realize that theoretically the percentage can take any real number, but typically one think of it as a number between 0% and 100%. That is why I said it might be misleading. Personally I found this kind of analogies more confusing than clarifying.