- #1

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100cos(θ) + 200cos(ω) = 250

100sin(θ) + 200sin(ω) = 0

How do you go about solving a system like this?

It's nonlinear, so linear algebra won't work...

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- #1

- 166

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100cos(θ) + 200cos(ω) = 250

100sin(θ) + 200sin(ω) = 0

How do you go about solving a system like this?

It's nonlinear, so linear algebra won't work...

- #2

chiro

Science Advisor

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For this particular problem, you will want to convert to a phase representation:

http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-rcostheta-alphaetc.pdf

- #3

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Thanks for that helpful link, but I don't think it applies to my system very much since I have two different angles. Am I wrong?

For this particular problem, you will want to convert to a phase representation:

http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-rcostheta-alphaetc.pdf

- #4

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I solved the system numerically, but I was wondering if any analytical solution exists.

- #5

chiro

Science Advisor

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In this case yes the numerical method is the best method. You are better off using a matrix method iteration scheme and you iterate until you get a change that is small enough from your last iteration.

- #6

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Algebra

We might try simplifying things by getting rid of the trig functions. In this case we can let ##\sin \theta = x## and ##\sin \omega = y##. Then ##\cos \theta = \sqrt{1 - x^2}## and ##\cos \omega = \sqrt{1 - y^2}##. With a little bit of massaging I believe you should get a quartic equation in one variable (or at any rate something like that) and can solve the whole thing.

Geometry

View the equations as components of an equation in vectors:

##100\begin{bmatrix}\cos \theta \\ \sin \theta\end{bmatrix} + 200\begin{bmatrix}\cos \omega \\ \sin \omega\end{bmatrix} = \begin{bmatrix}250 \\ 0 \end{bmatrix}.## Since the two vectors on the left hand side are unit vectors, this is really a geometry problem: you are asked to find the angles of a triangle whose side lengths are 100, 200, and 250 units. (You'll have to do a little work to figure out the signs of θ and ω.) Use your favorite techniques from analytic geometry (Law of Cosines).

- #7

- 3

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Square both equations.

Add the two and, after using sin

Solve for ω. and return to the original to solve for θ.

[I get cos ω = 92500 / 100000 = 0.925]

- #8

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cyto, how would I go about solving the general case? Can you explain that to me please? I'd prefer a nice, short, well-written explanation. I mean the case:

100cos(θ) + 200cos(ω) = 250

100sin(θ) + 200sin(ω) = 0

How do you go about solving a system like this?

It's nonlinear, so linear algebra won't work...

[tex]F=a[/tex]

[tex]G=b[/tex]

where F and G are some type of sine and cosine expressions. Well, what I'm getting at is that you really haven't solved this problem until you can. That's all.

Last edited:

- #9

verty

Homework Helper

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There is no general solution. Each problem is unique.cyto, how would I go about solving the general case? Can you explain that to me please? I'd prefer a nice, short, well-written explanation. I mean the case:

[tex]F=a[/tex]

[tex]G=b[/tex]

where F and G are some type of sine and cosine expressions. Well, what I'm getting at is that you really haven't solved this problem until you can. That's all.

- #10

- 4

- 3

Hello, jumping in a tad (sic...) late on this post but I'd like to ask what numerical method you would have used to solve this set of equation.I solved the system numerically, but I was wondering if any analytical solution exists.

I have derived the equations of equilibrium and geometric compatibility of a statically indeterminate system. I am left with a system of 5 equations and 5 unknown. The equations use trig functions and 3 different unknown angles.

- #11

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I have derived the equations of equilibrium and geometric compatibility of a statically indeterminate system. I am left with a system of 5 equations and 5 unknown. The equations use trig functions and 3 different unknown angles. I have tried to simplify the system using trig transformations but to no avail. I want to write an app to solve this set of equation. An analytical solution would have been great but I think I'll have to resort to numerical methods.

I later complexified my model a bit and ended up with a set of 14 trig equation with as many unknowns. So that one will definitely require a numerical method.

If some of you would actually like to see the physical model, I'd be happy to post a sketch of it. Maybe someone would have a better way of solving it. Let me know.

Thanks

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