Solving Equation from "Belt Problem": Find Alpha

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Discussion Overview

The discussion revolves around solving an equation derived from a "belt problem," specifically focusing on finding the variable alpha. The scope includes mathematical reasoning and problem-solving strategies related to transcendental equations.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Wernher presents an equation involving alpha and expresses uncertainty about its derivation and the next steps for solving it.
  • One participant suggests that due to the transcendental nature of the equation, an exact algebraic solution for alpha is unlikely, recommending a numerical approach instead.
  • Another participant questions the notation used in the equation, specifically the comma in "9,5," clarifying that it represents a decimal point in some countries.
  • A later reply reiterates the challenge of finding a neat solution, noting the periodic nature of the cosine function implies there are infinitely many solutions.
  • One participant proposes a graphical method to solve the equation by simplifying it and finding points of intersection between two functions.

Areas of Agreement / Disagreement

Participants generally agree that finding an exact solution for alpha is problematic due to the nature of the equation. Multiple approaches are suggested, including numerical and graphical methods, indicating a lack of consensus on a single solution path.

Contextual Notes

The discussion highlights the complexities of solving transcendental equations and the potential for multiple solutions due to the periodic properties of trigonometric functions. There are also considerations regarding notation that may affect interpretation.

itchy8me
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hi there. I have an equation i derived from a "belt problem" (i actually don't know if it's correctly derived yet). However i am now stuck and cannot find the next step to solving it, I'm trying to solve for alpha. The equation is:

\frac{1}{\alpha} * \left(\frac{4}{cos(\alpha)} + 9,5\right) = \frac{1}{36}

anybody know the direction i should take to solve this?

thanks,
wernher
 
Last edited:
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With the "unknown", \alpha both inside and outside a transcendental function, you aren't going to be able to find an exact, algebraic, solution. Your best bet is probably a numerical solution.
 
may i ask what is that comma in the equation?
 
darkmagic said:
may i ask what is that comma in the equation?

The equivalent of . (dot). In some countries the notation 9,5 is used instead of 9.5 (I know this to be the case in several European countries).


As far as the original equation goes you're not going to find a nice solution to it as HassofIvy mentioned. In fact due to the periodic nature of cos there are infinitely many solutions to the equation.
 
mmm.. i guess i'll have to go at the problem another way. thanks for the help, i probably would have stared at this for hours before moving on.
 
You could also solve it graphically. First simplify the equation to:

sec(α) = (1/144)α - 2.375

Then graph y1=sec(α) and the line y2=(1/144)α – 2.375

The points of intersection are solutions.
 

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