Question

x = 0.8*sinA-2.1*sinB / (1- cosB) as A/B=f(x)

Answer 1

>We need to know how much angle B changes at the change of A

This sounds like one of two things:

* the **rate of change** of B depending on A
   * called **dB/dA**
   * for example, when A=30° is increasing a tiny amount, then B is decreasing by 5 times this tiny amount
* the **increase** of B depending on A
   * called **ΔB**
   * for example, when A=30° is increasing to 35°, then B is decreasing by 110°

>certain length x

So x is a constant? You choose/measure/calculate it once, and then it keeps its value when A and B change, right?

Answer 2

First, mixing decimal numbers and algebra/calculus can be difficult to read.  So I'd recommend you either replace 0.8 and 2.1 with constants  (say a and b).   Or you convert them to fractions.   For instance x = (4sin(A)/5 - 21sin(B)/10)/(1-cos(B)) which can be rewritten as  x = (8sin(A)-21sin(B))/(10-10cos(B)).   

Beyond that, it's a bit unclear what you're saying regarding the relative rates of change of B and A.     If x is a fixed constant and A and B are related by x = (8sin(A)-21sin(B))/(10-10cos(B)) and you want to know the instantaneous rate of change of B with respect to A, then you'd take the derivative of that equation with respect to A and solve for dB/dA.   But that only makes sense if x is fixed.

Ignoring the relative rate of change question, you also said you want to rewrite x = (8sin(A)-21sin(B))/(10-10cos(B))  as A/B = f(x).    That would imply that given a value of x, it's possible to uniquely determine A/B.    The issue here is that this isn't true based only on what information you've provided.

For example we could have A/B = 10 by letting A=1 and B=0.1.   That corresponds to x being approximately 92.7826.    
  

  
But you can also have x \~ 92.7826 when B=0.05 and A\~0.28.  And in that case A/B = 5.6.
  

  
In other words, knowing that x = 92.7826 doesn't tell us anything about the quantity A/B.

Answer 3

From what i understood x is supposed to be some constant at the end and not a function, so we can find the rate of change of B with respect to A by the chain rule. If x is supposed to be a function then the rate of change with respect to A =0 unless they are dependent on a shared variable. So i will assume the first case, then dB/dA=
(-0.8•cos(A)/(1-cos(B))/(0.8•sin(A)•(-((1-cos(B))^-2 )) •sin(B)  -(2.1•(1/(cos(B)-1)))

Also wdym by A/B=f(x), if A and B are a functions of X then what is their expression in terms of X?