$3 \sin x + 4 \cos y = 5$, $4 \sin y + 3 \cos x = 2$ How to find $\sin x$, $\sin y$, $\cos x$, $\cos y$, 2020 contest question https://www.cemc.uwaterloo.ca/contests/past_contests/2020/2020CSMC.pdf
https://www.cemc.uwaterloo.ca/contests/past_contests/2020/2020CSMCSolution.pdf
Question 5 from 2020 CSMC math contest:
$$3 \sin x + 4 \cos y = 5,$$ $$4 \sin y + 3 \cos x = 2.$$
Find $\sin(x+y)$.
The solution page added that it is also possible to find $\sin x$, $\sin y$, $\cos x$, $\cos y$ with a different approach but I could not figure it out.

Hint: starting with $\sin x \cos y + \cos x \sin y=\frac{1}{6}$ and using that $\cos y=\frac{5-3\sin x}{4}$, $\sin y=\frac{2-3\cos x}{4}$ we can obtain
$$ \sin x \cdot (5-3\sin x)+\cos x \cdot (2-3\cos x)=\frac{2}{3}$$
$$ 5\sin x+2\cos x =\frac{11}{3} \implies \cos x=\frac{11}{6}-2.5\sin x $$We also have that $\cos^2 x+\sin^2x=1$ Substitute $\cos x$ and solve quadratic to find $\sin x$.
