Broken Calculator: only certain unary functions work. I have run into a challenge on Codecademy.com that has me absolutely bewildered. I'm sure I'm just overlooking an obvious solution, but I've been scouring tables of trigonometric and logarithmic identities for days and I still don't know where to start. The original task is here, and it's simply called "Broken Calculator".
I've found other "Broken Calculator" problems online, but this one is the first I've seen that makes use of trig and log functions.  Specifically, the only functions available are $e^x$, $ln$, $x^2$, √, $sin$, $cos$, $arcsin$, and $arccos$. The calculator defaults to 0, so you can get the number 1 by hitting $e^x$, and you can get 2 by hitting $e^x$ again, then $x^2  $, then $ln$. That's all simple.
Now, I can get any power of 2 by repeating the steps above, and similarly, given any number to start from, I can multiply or divide by a power of 2. That's easy.
The next challenge is to get 3, which I found using a 1, 2, √3 triangle and the appropriate trig functions:
$e^x$, $e^x$, √, $ln$, $arcsin$, $cos$, $e^x$, $x^2$, $ln$, $x^2$ -> 3
Again, I'm sure you find this trivial, but how would I use the same functions to get the number 11, or really any odd number for that matter other than 3?
Edit
The first two comments point out that the factorial operator is available for the first task where we need to find 3. However, in the very next task, we lose the factorial operator in exchange for the trig functions. I solved it using the steps above, but am stuck on the task to find 11.
 A: Indeed, all non-negative integers are achievable.
Operation 1: Using $e^x, x^2, \ln$, we can get from $y$ to $2y$.
Operation 2: Using $e^x, \sqrt, \ln$, we can get from $y$ to $\frac{y}{2}$.
Operation 3: Using $\sqrt, \arcsin, \cos, x^2$, we can get from $y$ to $1-y$, for $0<y<1$.
Clearly by operation 1, it suffices to achieve all odd positive integers. We now proceed by induction on $n \geq 1$ to prove that $2n-1$ is achievable.
When $n=1$, clearly $1$ is achievable using $e^x$.
Suppose that the statement holds for $1 \leq n \leq k, k \geq 1$. Consider $2(k+1)-1=2k+1$. Suppose that $2^a<2k+1<2^{a+1}, a \geq 1$. Then $1 \leq 2^{a+1}-(2k+1)<2^a \leq 2k-1$, and $2^{a+1}-(2k+1)$ is odd, so by the induction hypothesis $2^{a+1}-(2k+1)$ is achievable. We then start from this value, and apply operation 2 $a+1$ times to get $1-\frac{2k+1}{2^{a+1}}$, then apply operation 3 to get $\frac{2k+1}{2^{a+1}}$, then apply operation 1 $a+1$ times to get $2k+1$. Thus $2k+1$ is achievable, so we are done by induction.
I suppose a program could be written following the steps we undertook in the proof of the induction step...
A: This solution is admittedly ad hoc, but it works.
Generate $3$ via the triangle $1/2, \sqrt{3}/2, 1$ as you already accomplished. Now, generate $5$ via the triangle $\sqrt{3/8},\sqrt{5/8},1$; finally we generate $11$ through the triangle $\sqrt{5/16},\sqrt{11/16},1$.
The details are messy because one cannot define high-level intermediary operations. But this should definitely work.

In general, the triangles $\sqrt{p/2^n}, \sqrt{(2^n-p)/2^n},1$ will likely provide you with methods to generate every odd number. But this will undoubtedly be a messy circus.
A: Theorem. You can generate every odd positive integer.
Proof: Assume otherwise and let $n$ be the smallest positive odd number that cannot be generated.
Clearly, $n>1$ as
$$(0), e^x$$
produces $1$.
For $n>1$ odd, let $2^k$ be the smallest power of $2$ that is $>n$. Then $2^k<2n$ and $m:=2^k-n$ is a positive odd number $<n$, hence can be generated.
Then 
$$(m),e^x, \underbrace{\sqrt{x} , \ldots,\sqrt{x}}_k,\ln,\arccos,\sin,e^x,\underbrace{x^2,\ldots,x^2}_k,\ln $$
produces $n$. By simply concatenating the sequences thus found one gets a valid, but suboptimal result. Note that one may cancel any subsequences of the form $\ln, e^x$ or $\sqrt x,x^2$. $_\square$
