Why is the value of $\frac{\sin\theta}{\sin\sin\theta}$ close to the number of degrees in a radian? Why is $\cfrac{\sin\theta}{\sin\sin\theta}$ very close to 1 radian $(57.2958°)$? I tested this out for $\theta=1^{\circ}$ and $\theta=45^{\circ}$ and various other angles, but I always seem to get about the same answer.
 A: Presumably, you are calculating $\sin(\theta)$ with $\theta$ given in degrees. In that case, one has the following approximation of $\sin(\theta)$ for small $\theta$:
$$\sin(\theta)\approx \frac{\pi}{180}\cdot \theta$$
Since $\sin(\theta)$ always has absolute value no more than $1$, we can use this approximation to say that $\sin(\sin(\theta))\approx \frac{\pi}{180}\sin(\theta)$. Thus, one has
$$\frac{\sin(\theta)}{\sin(\sin(\theta))}\approx \frac{\sin(\theta)}{\frac{\pi}{180}\sin(\theta)}=\frac{180}{\pi}$$
which is what you are observing. Note that this expression never actually obtains equality, but stays pretty close.
It's worth noting that when you work in radians, you have that $\sin(\theta)\approx \theta$. So this term of $\frac{\pi}{180}$ comes in because, when the argument is in degrees, we need to multiply by $\frac{\pi}{180}$ to get the equivalent angle in radians. That is, the conversion factor shows up due to there being a conversion from degrees to radians implicit in the computation.
A: Consider $$y=\frac t{\sin(t)}$$ and, for small $t$ (using radians), by Taylor $$y=1+\frac{t^2}{6}+O\left(t^3\right)$$ Replace $t$ by $\sin(\theta)$ to get $$\frac{\sin(\theta)}{\sin(\sin(\theta)}=1+\frac 16\sin^2(\theta)+\cdots$$ which implies $$1\leq \frac{\sin(\theta)}{\sin(\sin(\theta)}\leq \frac 76$$
For the range $0\leq \theta \leq \pi$, the average value, given by $$\frac 1\pi \int_0^\pi\frac{\sin(\theta)}{\sin(\sin(\theta)}\,d\theta \approx1.09133$$ (this has been obtained using numerical integration) while $$\frac 1\pi \int_0^\pi\left(1+\frac 16\sin^2(\theta)\right)\,d\theta =\frac{13}{12}\approx 1.08333$$
A: When I put the expression into Wolfram Alpha I find that the function is not a constant. It has a minimum of 1 radian, as you thought, but a maximum of around 1.19 radians.
