How to Rearrange Trigonometric Equations for Integrals?

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In summary, the conversation discusses how to get something into integral form by rearranging the terms and using sum identities. The equation presented involves a limit as Δx approaches 0, which can be simplified by dropping some terms. However, it is noted that this may not result in something "times dx".
  • #1
Zula110100100
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To get something into integral form the dx needs to be at the end right? How can I do this if what I have is rsin(x+dx)cos(x+dx)?
 
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  • #2
It doesn't look ok. Can you perhaps write the whole thing/problem ?
 
  • #3
I am going to assume that your "dx" is not a differential but just some slight change in x. [itex]\Delta x[/itex] would be better notation.

The best you can do is use the sum identities:
sin(x+ dx)= sin(x)cos(dx)+ cos(x)sin(dx)
cos(x+ dx)= cos(x)cos(dx)- sin(x)sin(dx)

You could then argue that for dx sufficiently small sin(dx) is approximately equal to dx and cos(dx) is approximately equal to 1:
sin(x+ dx) is approximately sin(x)+ cos(x)dx
cos(x+ dx) is approximately cos(x)- sin(x)dx.

Now, r sin(x+ dx)cos(x+ dx) will be approximately
[itex]sin(x)cos(x)+ cos^2(x)dx- sin^2(dx)- sin(x)cos(x)dx^2[/itex]

I suppose you could not argue that, for dx a "differential", you can drop [itex]dx^2[/itex] to get [itex]sin(x)cos(x)+ cos^2(x)dx- sin^2(x)dx[/itex]

But those still will not give you something "times dx".


As dextercioby said, please show the entire problem.
 
  • #4
The entire equation is : Lim [itex]\Delta[/itex]x->0 [itex]\Sigma[/itex] (r[itex]\Pi[/itex][itex]\Delta[/itex]x/90 - rsin(xi)cos(xi)+rsin(xi+[itex]\Delta[/itex]x)cos(xi+[itex]\Delta[/itex]x))

Thats ugly as hell, but I think you get the idea yes?
 

What is the meaning of "Pulling dx out of cos(x+dx)?"

"Pulling dx out of cos(x+dx)" refers to a mathematical technique used in calculus to simplify expressions involving the cosine function and an infinitesimal change in the variable x, represented by dx. It involves using the fact that the cosine function is continuous and can be approximated by its value at a nearby point.

Why is it important to be able to pull dx out of cos(x+dx)?

This technique is important in calculus because it allows us to simplify complex expressions involving the cosine function and make them easier to work with. It also helps us to better understand the behavior of functions near a specific point.

What are the steps to pull dx out of cos(x+dx)?

The steps to pull dx out of cos(x+dx) are as follows:

  1. Expand the cosine function using the sum formula: cos(x+dx) = cos(x)cos(dx) - sin(x)sin(dx)
  2. Replace cos(dx) with 1 (since cos(dx) is approximately equal to 1 for small values of dx)
  3. Simplify the expression further by factoring out cos(x)
  4. Take the limit as dx approaches 0 to get the final result: cos(x+dx) = cos(x)

Are there any restrictions on when we can pull dx out of cos(x+dx)?

Yes, there are restrictions on when we can pull dx out of cos(x+dx). This technique only works when x is a variable and dx is an infinitesimal change in x. If x is a constant or dx is a finite value, then we cannot pull dx out of cos(x+dx).

What other functions can we pull dx out of in a similar way?

We can also pull dx out of other trigonometric functions such as sine and tangent, as well as exponential and logarithmic functions. However, the specific steps and restrictions may vary depending on the function. It is important to understand the underlying principles and rules of calculus in order to determine when and how we can pull dx out of other functions.

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