Cyrus the frog jumps $2$ units in a direction, then $2$ more in another direction. What is the probability that he lands less than $\sqrt2$ units from his starting position?
Placing the first jump along the $x$-axis to $(2,0)$ and the second at random angle $\alpha\in[0,2\pi)$, the squared distance from the start is $8+8\cos\alpha$.
Need $8+8\cos\alpha<2\Rightarrow\cos\alpha<-\frac34$; this holds on an arc of measure $2\arccos\frac34$ (using $\arccos(-x)=\pi-\arccos x$), so the probability is $\dfrac{\arccos(3/4)}\pi$.
Since $\arccos\frac34=2\arcsin\frac{\sqrt2}4$ (as $\cos\left(2\arcsin\frac{\sqrt2}4\right)=1-2\cdot\frac18=\frac34$), the probability is $\boxed{\dfrac{2\arcsin(\sqrt2/4)}\pi}$.
