Proving trigonometric identities in a belt and pulley proble

In summary, the conversation discusses how to verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C]. The solution involves drawing a sketch and using the relationship cos(θ)=(R-r)/c, where θ is 2 times the arc-cosine of [(D-d)/2C]. The conversation also emphasizes the importance of creating sketches when solving Physics or Geometry problems.
  • #1
Terrell
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Homework Statement


verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C]

2. The attempt at a solution
you can see my attempt in the second picture uploaded. i don't think i even got it right
 

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  • #2
Terrell said:

Homework Statement


verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C]

2. The attempt at a solution
you can see my attempt in the second picture uploaded. i don't think i even got it right
You are close, but you put A to the wrong place, and c is not the hypotenuse of a right triangle in your picture.
The line x (the connecting belt) makes a right angle with the radius of both circles, and you need to draw a parallel with it from the centre of the smaller circle. You get the yellow rectangle and the green right triangle. Find x and theta from that. Prove both formulas in the OP.
upload_2016-3-21_6-52-44.png
 
  • #3
ehild said:
You are close, but you put A to the wrong place, and c is not the hypotenuse of a right triangle in your picture.
The line x (the connecting belt) makes a right angle with the radius of both circles, and you need to draw a parallel with it from the centre of the smaller circle. You get the yellow rectangle and the green right triangle. Find x and theta from that. Prove both formulas in the OP.
View attachment 97665
wow! how did i not see that. thanks! 2(theta) equals to arccosine (D-d)/C right? so to further simplify... theta equals to arccosine[(D-d)/2C] did i got that right?
 
  • #4
Terrell said:
wow! how did i not see that. thanks! 2(theta) equals to arccosine (D-d)/C right? so to further simplify... theta equals to arccosine[(D-d)/2C] did i got that right?
Yes, cos(θ)=(R-r)/c, that is arccos((D-d)/(2c))=θ. But you also have to prove the other formula, for the length of the belt.
 
  • #5
one thing i learned is i really have to put all of my thoughts on paper to make things easier. thanks for the help i think i got! check the new image uploaded :)
 

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  • #6
Terrell said:
one thing i learned is i really have to put all of my thoughts on paper to make things easier. thanks for the help i think i got! check the new image uploaded :)
It is correct now. And you really need sketches. I am very old and have much experience, but still my first thing is to make a sketch before starting to solve a Physics or Geometry problem.
 
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  • #7
ehild said:
It is correct now. And you really need sketches. I am very old and have much experience, but still my first thing is to make a sketch before starting to solve a Physics or Geometry problem.
thank you a lot for responding to my thread! i will keep that in mind! :D best of luck
 

1. What is a belt and pulley problem in trigonometry?

A belt and pulley problem in trigonometry involves using trigonometric identities to calculate and solve for unknown angles or lengths in a system of pulleys and belts.

2. How do you approach solving a belt and pulley problem?

The first step is to draw a diagram of the system and label all known and unknown angles and lengths. Then, use trigonometric identities such as sine, cosine, and tangent to set up equations and solve for the unknown values.

3. What are some common trigonometric identities used in solving belt and pulley problems?

Some common identities include the Pythagorean identity (sin²θ + cos²θ = 1), the double angle identities (sin2θ = 2sinθcosθ and cos2θ = cos²θ - sin²θ), and the sum and difference identities (sin(θ±φ) = sinθcosφ ± cosθsinφ).

4. How can you verify that a solution to a belt and pulley problem is correct?

You can verify the solution by plugging the values back into the original equations and checking if they satisfy the given conditions. You can also use a graphing calculator to graph the equations and see if they intersect at the desired point.

5. Are there any tips or tricks for solving belt and pulley problems efficiently?

One tip is to start by simplifying the equations using trigonometric identities before plugging in values. It is also helpful to draw the system with the components in their correct positions to visualize the problem better. Additionally, practicing and familiarizing yourself with common trigonometric identities can make solving these problems easier and faster.

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