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I have this problem but I couldn't solve it. In a paper I'm reading for controlling a device, I need to generate the following angle

$$ \theta = \tan^{-1}\left( \frac{Y_{2} - Y_{1}}{ X_{2} - X_{1}} \right) $$ where $Y_{2} = 10, Y_{1} = 0, X_{2} = 10$ and $X_{1} = 0$.

Now I need to generate the following angle

$$ \phi = \sin^{-1} ( 0.401*\sin(\theta) - 2.208*\cos(\theta) ) $$

where $\theta = 0.7854$ (rad). The next angle $\psi$ is then generated as follows

$$ \psi = \sin^{-1} \left( \frac{0.401*\sin(\theta) + 2.208*\cos(\theta)}{\cos\phi} \right) $$

In my code, both angles $\phi$ and $\psi$ are undefined. I know that $\phi$ should be wrapped so that $\cos\phi \neq 0$ to avoid singularity but the problem $\phi$ is already undefined.

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1 Answer

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When $\theta=0.7854\approx \frac{\pi}{4}$, $\sin{\theta}=\cos{\theta}=\frac{\sqrt{2}}{2}$. Evaluating your definition for $\phi$ at $\frac{\pi}{4}$ comes out to $-1.278$, which is not in the domain of $\sin^{-1}{\theta}$ (because it is not in the range of $\sin{\theta}$). The problem seems to be in the part of your expression for $\phi$ that is inside the inverse sine function, since it is generating values that are not in $[-1,1]$.

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