Limit of $\frac{\sin(\theta)}{\theta}$ in degrees What does $\lim \limits_{\theta\to0}\dfrac{\sin(\theta)}{\theta}$ equal when $\theta$ is expressed in degrees?

I know that theta in degrees is $\frac{\pi}{180}$ theta radians, but I don't get the final answer of $0.01745$.
$$
\lim_{\theta\to0}\frac{\sin\theta^\circ}{\theta^\circ} = \lim_{\theta\to0} \frac{\sin\left( \dfrac{\pi\theta}{180}\text{ radians} \right)}{\theta} = \lim_{\eta\to0} \frac{\sin(\eta\text{ radians})}{\left( \dfrac{180\eta}{\pi} \right)} = \frac\pi{180}\lim_{\eta\to0} \frac{\sin\eta}{\eta}.
$$

This is why radians are used: When radians are used then $\lim\limits_{\eta\to0} \dfrac{\sin \eta} \eta=1$.

Postscript in response to comments below:

The question is how to find
\begin{align}
&amp; \lim_{\theta\to0} \frac{\text{sine of $\theta$ degrees}}\theta = \lim_{\theta\to0}\frac{\text{sine function in radians}\left(\dfrac{\pi\theta}{180}\right)} \theta \\[12pt]
= {} &amp; \frac\pi{180}\lim_{\theta\to0} \frac{\text{sine function in radians}\left(\dfrac{\pi\theta}{180}\right)} {\dfrac{\pi\theta}{180}} = \frac\pi{180}\lim_{\eta\to0} \frac{\text{sine function in radians}(\eta)}\eta.
\end{align}

In other words the notation "$\sin$" means a particular function: the one that maps a number $\eta$ to the sine of $\eta$ radians.

End of postscript

It's the same as the reason why $e$ is the "natural" base for exponential functions:
\begin{align}
\frac{d}{dx} 2^x &amp; = (2^x\cdot\text{constant}) \\[10pt]
\frac{d}{dx} 3^x &amp; = (3^x\cdot\text{a different consant}) \\[10pt]
\frac{d}{dx} 20^x &amp; = (20^x\cdot\text{yet another constant})
\end{align}
etc.  Only when the base is $e$ is the "constant" equal to $1$.
The question is not properly posed.
In whatever units the is angle chosen the result is the same $(=1).$
For all units (degree, radian)
$$ lim_{X \rightarrow 0}\dfrac{\sin X}{X}$$
equals unity.
You should get that answer independently  when asked &quot; how many radians are there in a degree?&quot;