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$$(x+a \cos\theta)^2+(y-a \sin\theta)^2=b^2$$

expanding and using the Weierstrass substitution we find that

$$\theta= 2 \arctan \frac{\left( 2ay- \sqrt{ 4a^2y^2 - ( (x-a)^2+y^2-b^2)( (x+a)^2+y^2-b^2) }\right)}{(x-a)^2+y^2-b^2} $$

if we use the law of cosines, with $c^2=x^2+y^2$

$$\theta = \arctan_2(y,x) - \arccos( (c^2+a^2-b^2) / (2ac) )$$

Is there a way to pass from one expression to the other using trigonometric identities?

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What do you mean by arctan2(y,x) – Amr Nov 13 '12 at 16:44
@Amr – Henry Nov 13 '12 at 16:45
an extended version of the tangent that returns the correct angle (between -pi and pi instead of -pi/2.. pi/2) considering the signs of y and x – Fabio Dalla Libera Nov 13 '12 at 16:47
up vote 2 down vote accepted

Try this: $$(x+a \cos\theta)^2+(y-a \sin\theta)^2=b^2$$ Working through we get $$x\cos\theta-y\sin\theta=\frac {(b^2-a^2-y^2-x^2)}{2a}$$

Now use $x^2+y^2=c^2$ on both sides and set $\phi=\arctan \frac y x$. Divide through by $c$ to obtain: $$\cos \phi \cos \theta - \sin\phi\sin\theta =\cos(\theta+\phi)=\frac {(b^2-a^2-c^2)}{2ac}$$ and you can do it from there.

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Thank you, you're completely right! I wonder why Mathematica cannot compute (symbolically) that the difference between the two quantities is 0 :) – Fabio Dalla Libera Nov 14 '12 at 15:38

You could look at the angles at the vertex between sides $c$ and $a$. The arctan comes from the $x,y$ triangle and the arccos from the $abc$ one.

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This is how I derived the two equations. I would like to pass from one to the other using trigonometric identities, i.e. manipulating the equations. Imagine the first theta is called theta1, and the second is theta2. I want to show, with things like that they are the same – Fabio Dalla Libera Nov 13 '12 at 17:19
Sorry, I had misunderstood. – Mark Bennet Nov 13 '12 at 17:25

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