Joint pdf of functions $Y$ and $Z,$ both of which depend on uniformly distributed $X$

Question

This is my first time using the math stack, so please bear with me!

I have read the textbook over and over and I cannot for the life of me figure this out.

Let $X$ be a uniformly distributed random variable on $[0, 2\pi].$ You also have $Y=\sin(X)$ and $Z=\cos(X),$ and $X, Y, Z$ are all continuous rvs. How do you find the joint pdf of $Y$ and $Z? $

Since $Y$ and $Z$ depend on $X,$ and due to the nature of their respective functions, their ranges are also the same as $X.$ So I came up with:

$$p(y,z) =\int_0^{2\pi} \int_0^{2\pi} f(y,z) \,dzdy$$

(I apologize for the lack of formatting, I couldn't figure out how to do it).

How do I proceed from here?

Answer 1

Comment: I think it is important for you to visualize the support of the joint distribution of $Y$ and $Z.$ I sampled 100 points $X_i \sim \mathsf{Unif}(0, 2\pi),$ independently, using R statistical software and then plotted the results.

x = runif(100, 0, 2*pi)
y = sin(x);  z = cos(x)
plot(y,z, pch=20)

How do you account for this result? This figure is certainly not a proof of anything, but do you suppose the joint distribution could be uniform on the unit circle? How could you write the joint density function? How about making some use of polar coordinates?

Here is a 'stripchart' showing the values of $X.$