Consider the independent random variables $U_1 =U(0,c)$ $U_2=U(0,\pi/2)$
The challenge is to compute the probability:
$$P(U_1 \leq a\text{cos}(U_2))$$
An attempt:
$$P(U_1 \leq a\text{cos}(U_2))=P(U_1-acos(U_2)\leq0)$$
Applying convolution: $$\int_{-\infty}^0\int_{-\infty}^{\infty} f_{U_1}(t)f_{-a\text{cos}(U_2)}(x-t)dtdx$$ By independence: $$=\int_{-\infty}^0\int_{-\infty}^{\infty} f_{U_1,-a\text{cos}(U_2)}(t,x-t)dtdx$$
This is where I get stuck, I believe using multivariate change of variable does the trick. The transform should give:
$$=\int_0^{\pi/2}\int_0^{a \text{cos(t)}} f_{U_1,U_2}(x,t)dxdt$$
From which it's no trouble to carry out the computation to the end. Is it correct to use multivariable change of variable in the above, and if so, what transformation should be performed?