I made 2 equations but can they become 1

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Discussion Overview

The discussion revolves around the possibility of combining two equations related to a CNC machine operation involving a radius and chord. The focus is on whether these equations can be simplified into a single expression for practical application.

Discussion Character

  • Exploratory, Technical explanation

Main Points Raised

  • One participant describes the need for two equations to manipulate a CNC machine for a specific operation involving a radius equation.
  • Another participant questions the meaning of "Asin" in the first equation, asking if it refers to arcsine or a multiplication by a constant.
  • A later reply clarifies that "Asin" is interpreted as arcsine.
  • One participant reformulates the equations using simplified notation, expressing the first equation as \(x = \arcsin(C/r)\) and substituting this into the second equation to yield \(y = r\sin(0.5\arcsin(C/r))\).
  • A subsequent response expresses gratitude for the simplification, indicating it will facilitate calculations on a specific calculator model.

Areas of Agreement / Disagreement

Participants generally agree on the interpretation of the equations and the simplification process, but there is no explicit consensus on whether the combined equation fully meets the original intent of the problem.

Contextual Notes

The discussion does not address potential limitations or assumptions in the equations, nor does it explore the implications of the combined equation in practical scenarios.

DRMSquared
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So a little background then to the point, I needed to get a cnc machine to do something it don’t want to do, so I came up with 2 equations to lie to the machine to get it to do an operation correctly. It is a radius equation that finds the center point of a radius if it is not a full radius. Can these be combined to make one?

1st: Asin(chord/radius)=x
2nd: radius[sin(x/2)]=y

Y is the input for the machine, it’s the important answer.
 
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Welcome, DRMSquared! (Wave)

DRMSquared said:
Can these be combined to make one?

1st: Asin(chord/radius)=x
2nd: radius[sin(x/2)]=y

Y is the input for the machine, it’s the important answer.
In the first equation, does Asin mean the Arcsine, or does it mean $A$ times sin where $A$ is some constant?
 
Euge said:
Welcome, DRMSquared! (Wave)

In the first equation, does Asin mean the Arcsine, or does it mean $A$ times sin where $A$ is some constant?

It would be sin-1 so I’m guessing arcsine
 
Ok. For simplicity, let $C = \text{chord}$ and $r = \text{radius}$. Equation 1 is then $x = \arcsin(C/r)$. Plugging in the expression of $x$ into Equation 2 yields $$y = r\sin(.5\arcsin(C/r))$$
 
Euge said:
Ok. For simplicity, let $C = \text{chord}$ and $r = \text{radius}$. Equation 1 is then $x = \arcsin(C/r)$. Plugging in the expression of $x$ into Equation 2 yields $$y = r\sin(.5\arcsin(C/r))$$

You my friend are awesome, thank you so much, now I can easily plug this into my hp48g and have an easier and quicker time doing this problem.
 

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