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Homework Statement
Solve for ##\theta##:
##\cot \theta \sin \beta + \rho \csc^2 \theta = \cos \beta##
where ##0^\circ<\beta<90^\circ, \ 0^\circ<\theta<90^\circ##, and ##0<\rho<1##.
Homework Equations
##\cot^2 x +1 = \csc^2x##, the quadratic formula.
The Attempt at a Solution
##\cot \theta = \frac{\sin \beta}{2 \rho} \pm \sqrt{\Big(\frac{\sin \beta}{2 \rho}\Big)^2 + \frac{\cos \beta}{ \rho}-1}##
This is not really a homework question, but since it looks like one I decided to post it here. It's actually a relationship between the variables involved in grinding an edge tool such as a carpenter's chisel. ##\rho## is the ratio of the tool's thickness to the diameter of the circular grindstone, so it's a positive constant, usually much smaller than unity. The issue is that a round grindstone produces a hollow grind rather than the flat surface you'd get sharpening on a flat stone, and I'm interested in the geometry of the hollow grind. ##\beta## is the bevel angle, or angle that the two edges make when they meet at the cutting edge.
##\theta## is the angle the two edges would make if the grinding were done on a flat surface.
My solution seems to work as it gives results that match what I measure and also that a friend got using AutoCAD or some such software program. There are two roots to the solution, though. The positive root gives the value of ##\theta## but the negative root gives the supplement of ##\theta-\beta##. I realize this negative root is outside the bounds of ##\theta## but it is nevertheless a curiosity to me.
So I have two questions. Can anyone derive a simpler solution and can anyone explain the negative root's value?