A confusion regarding circular functions of real numbers

Question

I recently started $11^{th}$ grade Trigonometry.

Prior to this, I learnt about the three general systems of angle measurement, mainly the Circular System of Angles, dealing with radians

The very first section in the first chapter related to Trigonometry was regarding Trigonometric Functions of Real Numbers.

Shouldn't Trigonometric Functions have angles as inputs, in degrees, grades, radians or some other angle unit.

I did read about the relationship between real numbers and radian measures. I think I misunderstood some part of it.

It doesn't mean that $x^c = x$, right? In my opinion, it means that if $x^c$ is some angle, then $x \in \Bbb R$. Which one of the above mentioned assumptions is right?

I also am confused about the term radian measure. In my opinion, it means that if $\theta = x^c$, then $x$ is called the radian measure of $\theta$. Am I right here as well?

If I'm right in both of these cases, does the term trigonometric functions of real numbers mean that $sin$ $x$ is a way of saying $sin(x^c)$, since the superscript $^c$ is usually omitted while writing radians?

This might sound like a simple question but its really confusing for me. Please let me know about any misconceptions I might have about radians and also what the right thing is.

Thanks!

Answer 1

Radians and degrees are two different units of angle measure. Like we measure length in metres or time in seconds, we measure angles in radians. Trigonometric functions do take angles as inputs in both units. So, $\sin 60^o , \sin \frac{\pi}{3}$ are both equivalent, just as $100\ cm$ is equivalent to a meter.

Answer 2

An arc of a radius-$r$ circle subtending an angle $\theta$ at its centre has length $s=kr\theta$ for some constant $k$, which depends on the units in which you measure angles. Regardless of that choice, $\theta$ being a dimensionless number implies $k$ is too. If we work with degrees, $k=\frac{\pi}{180}$; if we work with radians, $k=1$, which is the motive for using radians. So now $s=r\theta$, angles are dimensionless, and "$1$ radian" is just a fancy name for $1$. Why we even mention radians given this fact is another question, which I've addressed before. But if you bear this point in mind, all your concerns evaporate. For example, $\sin0.3$ is synonymous with $\sin(0.3\,\text{radians})$, because $0.3$ is synonymous with $0.3\,\text{radians}$.