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: 
You can write $E$ as

$$E=f^{-1}[\{C\}]\cap(I\times\Bbb R^2)\;,$$

which makes it clear that $E$ is a closed subset of $\Bbb R^3$; thus, $E$ is compact if and only if it is bounded in $\Bbb R^3$.
In general, no, it is not compact. If, say, $f$ is the constant function $C$, then $E=I\times\Bbb R^2$.

However, if $f$ is a [proper map](https://en.wikipedia.org/wiki/Proper_map), then, yes, $E$ will be compact.