I'm trying to answer a two part problem.
I've already shown that if $X, Y$ are independent standard normal random variables with $R, \theta$ defined by $X=R\cos\theta$, $Y=R\sin\theta$, then the joint distribution of $R$ and $\theta$ is $$ f(r,\theta)=\frac{r}{2\pi}e^{-\frac{r^2}{2}} $$
The next part of the question asks me to let $X$ and $Y$ be independent standard normal random variables, and to use the above to show that for a constant $c \gt 0$,
$$ \mathbb{P}(X \gt 0, Y \gt -cX) = \frac{1}{4} + \frac{\tan^{-1}(c)}{2\pi} .$$
I've really got no idea where to begin with this, so I'd really appreciate it if someone could show me where to get started.
Theorem. If $A$ is a continuous random variable in $\mathbb{R}^n$ with density $f$, $S$ is an open subset of $\mathbb{R}^n$ such that $P(A\in S)=1$, $\varphi:S\mapsto\mathbb{R}^n $ is a $C^1$ function and a bijection, $J_{\varphi}\neq0$, then density of $B=\varphi(A)$ is of the form: $$g(a)=f(\varphi^{-1}(a))|J_{\varphi^{-1}}(a)|,\,a\in\varphi(S) .$$
Let $A=(X, Y)$, $\varphi(x, y)=(x, y+xc)$ and $B=(X, Y+cX)$, and use the theorem.
We know what density of $A$ is, because since $X$ and $Y$ are independent, we have $f_{(X, Y)}(x, y) = f_X(x)f_Y(y)$.
What is left, is to calculate $$\int\limits_0^\infty\int\limits_0^\infty f_{(X, Y+cX)}(x, y)\,\text{dx}\,\text{dy} $$