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I've been slaving over this for a couple days now and I can't seem to get a resolution at all.
Can anyone solve for $\alpha$? example

Given data is the length of the red line that isn't tangential, $\angle$ $\delta$, $\angle$L and The length of the radius of the circle.

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What is the data given on this problem ? –  Belgi Sep 27 '12 at 0:54
    
The given data is $\partial$ and L. I probably shouldn't have used $\partial$ as a variable :S –  Korvin Szanto Sep 27 '12 at 0:57
    
So, the radius of the circle is not given? –  Gerry Myerson Sep 27 '12 at 1:05
    
I misspoke, Radius of the circle is given aswell, I've edited my question. –  Korvin Szanto Sep 27 '12 at 1:05
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1 Answer

up vote 2 down vote accepted

Since you have the lengths and directions of two sides of the triangle, you can compute the coordinates of its right-hand vertex relative to the centre of the circle; their ratio gives you $\tan\alpha$:

$$ \alpha=\arctan\frac{r\sin\partial+d\sin(\partial-\pi+L)}{r\cos\partial+d\cos(\partial-\pi+L)}\;, $$

where $r$ is the radius and $d$ is the length of the red line.

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This solution is so utterly simple, but it baffled me for days, thank you very much! –  Korvin Szanto Sep 27 '12 at 1:11
    
@KorvinSzanto, the solution is clever, not so simple though! –  Emmad Kareem Sep 27 '12 at 1:19
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