# Math Help - sin a = t.a

1. ## sin a = t.a

I'm trying to solve the equation..

sin a = t a ...or rewritten as... sin a - t a = 0

solve to find 'a' for a given value of t
where a is in radians, 0 < a < pi/2
where t < 1, typically 0.7-0.9.

...Problem background...

Taking a slice of pizza, measure around the arc of the crust, and then measure the straight chord line across the crust. Given these 2 measurements, it is possible to find the slice angle of the pizza, the angle, etc.

c = arc length
l = chord length

So taking a as the unknown angle of the slice, and let t = l/c, which is a known constant, the solution reduces down to

sin (ang/2) = t . (ang/2)

or by making a = ang/2, the equation can be simplified to

sin a = t.a ...or rewritten as... sin a - t.a = 0

(a is measured in radians)
where (2/pi < t < 1) and (0 < a < pi/2)

However, I can only solve this by trial and error, using iterative techniques, such as Newton iteration. It would be nice if there was a precise solution for the above equation to solve for a.

Anyone got a solution, or is this equation not possible to solve?

2. Originally Posted by surfdabbler
I'm trying to solve the equation..

sin a = t a ...or rewritten as... sin a - t a = 0

solve to find 'a' for a given value of t
where a is in radians, 0 < a < pi/2
where t < 1, typically 0.7-0.9.

...Problem background...

Taking a slice of pizza, measure around the arc of the crust, and then measure the straight chord line across the crust. Given these 2 measurements, it is possible to find the slice angle of the pizza, the angle, etc.

c = arc length
l = chord length

So taking a as the unknown angle of the slice, and let t = l/c, which is a known constant, the solution reduces down to

sin (ang/2) = t . (ang/2)

or by making a = ang/2, the equation can be simplified to

sin a = t.a ...or rewritten as... sin a - t.a = 0

(a is measured in radians)
where (2/pi < t < 1) and (0 < a < pi/2)

However, I can only solve this by trial and error, using iterative techniques, such as Newton iteration. It would be nice if there was a precise solution for the above equation to solve for a.

Anyone got a solution, or is this equation not possible to solve?
Newton iteration is the best way to solve this. In theory you could use the Taylor series for sin(x) and truncate it at a suitable point but remember it'll only work for 0 < |x| < 1

3. Ok, I suspected this might be the case. Thanks!