Find Lambda from this arbitrary triangle

In summary, the conversation discusses obtaining equation 3.2.31 based on a given schematic, specifically focusing on finding the angles and magnitudes of complex schematics. The individuals discuss using trigonometric functions and manipulating equations to visualize and solve the problem. Through their discussion, they arrive at a solution for lambda by extending the side length a to create a right triangle with hypotenuse p. This conversation highlights the use of trigonometry in solving complex schematic problems.
  • #1
bugatti79
794
1
Folks,

I am puzzled how one obtains equation 3.2.31 based on the schematic as attached! Can you help?
Is there an online source I can refer to to learn how to obtain angles and magnitudes of complex schematics?

Thanks
 

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  • #2
I am still working on this one...
However, I feel like the secret will be in manipulating the equation you have.
Edited*
##\frac{\sin \psi }{ m - \cos \psi } = \frac{\sin( \pi - \psi) }{ a/p +\cos( \pi - \psi) } = \frac{p\sin( \pi - \psi) }{ a + p \cos ( \pi - \psi) }##
This should help you visualize the triangle you should be working with for ##\tan \lambda##.
 
Last edited:
  • #3
Hi, thanks...
This is as far as I got...see attached...
 

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  • #4
That looks good. You extension works to give you a solution for lambda. But it doesn't help get to that tan lambda formula you were looking for.

Try to extend the side length a to make a right triangle with hypotenuse of p. Use sin and cos to determine the side lengths.
Then you can directly get the formula for tangent using the opposite/adjacent of the appropriate right trangle which includes angle lambda.
 
  • #5
Hi RUber,

I have it. The extended segment of "a" which forms a right angled triangle with P is Pcos(pi-psi). The tan of lambda is obvious. Thank you very much.
I have learned a new way of tacking triangles!
 
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Likes RUber

1. What is Lambda in relation to a triangle?

Lambda, or Λ, is a symbol used to represent the ratio of the area of a triangle to the area of its circumscribed circle. It is also commonly used to represent the ratio of the length of a side of a triangle to the radius of its inscribed circle.

2. How do you calculate Lambda from a given triangle?

To calculate Lambda, you will need to know the length of at least one side of the triangle and the radius of its circumscribed or inscribed circle. Then, you can use the formula Λ = (Area of triangle)/(Area of circumscribed/inscribed circle) or Λ = (Length of side)/(Radius of circumscribed/inscribed circle).

3. Can Lambda be negative or zero?

No, Lambda cannot be negative or zero. It represents a ratio of areas or lengths, and therefore must be a positive value.

4. What is the significance of Lambda in geometry?

Lambda is important in geometry as it helps us understand the relationship between the sides and angles of a triangle and the properties of its circumscribed and inscribed circles. It is also used in various geometric proofs and constructions.

5. Are there any other uses for Lambda in mathematics?

Yes, Lambda is also commonly used in calculus, statistics, and other fields of mathematics. In calculus, it represents a variable in a function or equation. In statistics, it is used to denote the rate of decay in a Poisson distribution. It also has various other applications in physics, engineering, and economics.

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