If $X=\sin\Theta$ and $Y=\cos\Theta$ with $\Theta$ uniformly distributed, how can I compute the joint pdf of $(X,Y)$? I have a random variable $\Theta$ uniformly distributed between $[-\pi ,\pi]$, two functions $X=\sin\Theta$ and $Y=\cos\Theta$. I know that $X$ and $Y$ are uncorrelated but not independent. I want to find the joint pdf $f(x,y)$ of $X$ and $Y$. How can I compute this?
I tried by using cdf $F(x,y)$, defined as:
\begin{align}
F(x,y)&=P(X<x,Y<y)
\\&=P(\sin\Theta<x,\cos\Theta<y)
\\&=P(\Theta<\arcsin x,\Theta<\arccos y)
\\&=P(\Theta<\max(\arcsin x,\arccos y))
\end{align}
(Is the last equality right?)
So:
\begin{align}
F(x,y) = \begin{cases}c_1\arccos x+c_2&,\text{ if }x,y\le \frac{\sqrt 2}{2} \\ c_3\arcsin y+c_4 &,\text{ if }\frac{\sqrt 2}{2}\le x,y\le 1\end{cases}
\end{align}
By imposing the property of cdf (i.e. $F(-1,-1)=0$ and $F(1,1)=1$):
\begin{align}
F(x,y) = \begin{cases}-\frac{2}{3\pi}\arccos x+ \frac23 &,\text{ if }x,y\le \frac{\sqrt 2}{2} 
\\ \frac2{\pi}\arcsin y &,\text{ if }\frac{\sqrt 2}{2}\le x,y\le 1\end{cases}
\end{align}
Now, I want to find the joint pdf $f(x,y)$ as:
$$f(x,y) = \frac{\partial ^2 F(x,y)}{\partial x\partial y}$$
How can I proceed (if the procedure that I used is correct)?
If the my procedure isn't right, how do I calculate the joint PDF of $(X,Y)$? 
Thank you in advance!
 A: We can derive a CDF, but not a valid pdf, as pointed out by @whuber. I will demonstrate how to derive the CDF. 
You are correct up until here:
$$\eqalign{
F(x,y) &= P(X \leq x, Y \leq y) =  P (\sin(\theta) \lt x, \cos(\theta)\lt y) \\
&= P(\theta \leq \arcsin(x), \theta \leq \arcsin(y)).}$$
However, in your next step, you write $\max$ where you should have $\min$ (since $\theta$ must be less than both, it must be less than the smaller of the two). Therefore, we have 
$$F(x,y) = P\left(\theta \leq \min\{\arcsin(x), \arcsin(y)\}\right).$$
Since $\theta \sim U(-\pi, \pi)$, it follows that 
$$F(x,y) = 
\begin{cases} 
      0, & \min\{\arcsin(x),\, \arcsin(y)\} \leq -\pi \\
      \frac{\min\{\arcsin(x),\ \arcsin(y)\} + \pi}{2\pi}, & -\pi \leq \min\{\arcsin(x),\, \arcsin(y)\}\leq \pi \\
      1, & \pi \leq \min\{\arcsin(x),\, \arcsin(y)\} 
   \end{cases}
$$
A: *

*$\Theta \sim U(-\pi, \pi)$ so the density of $\Theta$ is given by $\frac{1}{2 \pi}$ in $-\pi, \pi$.


*$F(x, y) = P(X \le x, Y \le y) = P(\Theta \le \arcsin(x) \wedge \arccos(y))$.


*$
F(x, y) = \frac{1}{2\pi}\int_{-\pi}^{\arcsin(x) \wedge \arccos(y)} 1 \cdot d\theta
= \frac{1}{2\pi}\arcsin(x) \wedge \arccos(y) + \frac{1}{2}.
$
