Help with extracting alpha from: -alpha-asin(sin(alpha)(r/l)

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In summary, the conversation is about using the Law of Cosines to extract an angle from an equation involving a shaft turning on an air engine. The person is struggling with finding a way to solve for alpha and is concerned about the (r/l) term in the equation. They mention that arcsin(sin(x))=x but are unsure if this applies to their equation. The conversation concludes with a note about the arcsin being undefined for certain values of alpha.
  • #1
dbarger1225
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I am using the Law of Cosines to extract and angle I need from a shaft that is turning on an air engine by piston oscillation and I am having a brain fart on if there are any identities I am missing that can help me pull '[itex]\alpha[/itex]' from the right side of my equation:

acos(([itex](h-pl)^{2}[/itex]-[itex]r^{2}[/itex]-[itex]l^{2}[/itex])/(-2*r*l))-[itex]\pi[/itex]=-[itex]\alpha[/itex]-asin(sin([itex]\alpha[/itex])(r/l))

I would like to solve the entire equation for [itex]\alpha[/itex] and the other variables can be treated like constants.

Thanks for the help or attempts in advance.
 
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  • #2
well - you didn't close the brackets in the arcsine, so it's hard to tell - but asin(sin(x))=x
 
  • #3
I fixed the bracket issue, I apologize about that.

I am concerned with the (r/l) term inside of the equation. I understand that asin(sin(x))=x .. but I'm certain that asin(sin(x)(r/l)) --DNE-- (r/l)x
 
  • #4
Oh I think I see, it wasn't clear to me if the r/l was multiplied with the alpha or the sine-alpha, or with the arcsine.
This what you mean?

[tex]\sin^{-1}( \frac{r}{l}\sin\alpha )[/tex]

... afaik you can't extract alpha from this sort of equation - you need to be cleverer in the setup or use an approximation (or use a numerical method).

Equation is of form: x+sin-1(A.sin(x))=b ... solve for x. argh. (assuming the RHS is all constants)

Note: the arcsin will be undefined for some alpha, depending on the value of rl - you want |sin(α)| < l/r (something...)

Where did you start from?
 
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  • #5


Hello,

Thank you for reaching out for help with your equation. It seems like you are trying to use the Law of Cosines to extract the angle \alpha from your equation. However, the equation you have provided is not solvable for \alpha using only the Law of Cosines. This is because the equation contains both trigonometric functions of \alpha (sin and asin) and also contains a constant (\pi). In order to solve for \alpha, you will need to use a different approach.

One possible way to solve for \alpha is to use the identity sin^2(\alpha) + cos^2(\alpha) = 1 to eliminate the sin function in your equation. This will give you a quadratic equation in terms of cos(\alpha), which you can then solve for \alpha using the quadratic formula. Another approach is to use the identity cos(\alpha) = sin(\alpha + \pi/2) to rewrite your equation in terms of only cos(\alpha), which can then be solved using the Law of Cosines.

I would also recommend checking your equation and making sure all the variables and constants are correctly represented. It is possible that there may be a typo or missing term that is causing difficulties in solving for \alpha.

I hope this helps and good luck with your calculations!
 

1. What is alpha in the given equation?

Alpha is a variable that represents an angle in the given equation.

2. How do you extract alpha from the equation?

To extract alpha, you can use inverse trigonometric functions such as arcsine (asin) or inverse sine (sin^-1) on both sides of the equation.

3. What does the (r/l) term represent in the equation?

The (r/l) term represents the ratio of the radius (r) to the length (l) of a given object or shape.

4. Can this equation be used to find the value of alpha in any situation?

No, this equation is specific to finding the value of alpha in a scenario where the radius and length of an object are known.

5. What are some real-life applications of this equation?

This equation can be used in various fields such as engineering, physics, and geometry to calculate angles in different shapes and structures.

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