<s>[INST] How does $\sqrt{y^2 + y^2} = 1$ give $y = -1/\sqrt{2}$ here? I'm studying this problem:

and I cannot see the last step, how we go from $$\sqrt{y^2 + y^2} = 1$$ to $$y = -1/\sqrt{2}$$
When I try to solve this, I end up with:
$$y^2 + y^2 = 1^2 = 1$$
$$2y^2 = 1$$
$$y^2 = 2$$
$$y = \sqrt{2}$$
 [/INST] There are basically two interpretations here: either solve $\sqrt{y^2+y^2}=1$ to get $-\frac{1}{\sqrt{2}}$.
Or, the other interpretation is: They didn't need to solve the equation $\sqrt{y^2+y^2}=1$. They just made the observation that $$\cos\left(\frac{5\pi}{4}\right)=\sin\left(\frac{5\pi}{4}\right)=-\frac{1}{\sqrt{2}}$$
To see this, recall that the sine or cosine of $45^{\circ}$ or $\frac{\pi}{4}$ is $\frac{1}{\sqrt{2}}$. 
So $\sin\left(\frac{5\pi}{4}\right)=\sin\left(\frac{\pi}{4}+\pi\right)$ (a translation in the negative $x$ direction).
By 'they', I mean the person who wrote the paper.
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