Solving systems of equations with trig functions

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Discussion Overview

The discussion revolves around solving a system of nonlinear equations involving trigonometric functions. Participants explore various methods for finding solutions, including numerical and analytical approaches, while addressing the complexities introduced by multiple angles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a system of equations involving cosine and sine functions and expresses difficulty in solving it due to its nonlinear nature.
  • Another suggests converting to a phase representation but acknowledges that it may not apply due to the presence of different angles.
  • A participant claims to have solved the system numerically and inquires about the existence of an analytical solution.
  • One response proposes two potential analytical approaches: one using algebraic manipulation to eliminate trigonometric functions and another using geometric interpretation of the equations as vector components.
  • Another participant suggests isolating terms and squaring the equations to derive a solution for one of the angles.
  • There is a request for a general method to solve similar systems, but a participant asserts that no general solution exists as each problem is unique.
  • Several participants discuss their experiences with numerical methods and express interest in sharing more complex systems involving multiple equations and unknowns.

Areas of Agreement / Disagreement

Participants generally agree on the challenges posed by the nonlinear nature of the equations and the potential for both numerical and analytical solutions. However, there is no consensus on the best approach, and multiple competing views on the feasibility of analytical solutions remain.

Contextual Notes

Some participants mention the complexity of their systems, including multiple unknown angles and the potential need for numerical methods, indicating that the discussion may be limited by the specific forms of the equations presented.

Who May Find This Useful

This discussion may be useful for individuals interested in solving nonlinear systems of equations, particularly those involving trigonometric functions, as well as those exploring numerical methods and analytical approaches in mathematical modeling.

cytochrome
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I've stumbled upon a system of equations that involves trig functions...

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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I solved the system numerically, but I was wondering if any analytical solution exists.
 
Sorry I thought your angles were on the same line.

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.
 
An analytic solution is possible. Here are two potential approaches. One uses algebra only. The other uses geometry.

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).
 
Just isolate 100 cos θ in one equation, and 100 sin θ in the other equation.
Square both equations.
Add the two and, after using sin2+cos2=1, you are left with an equation involving cos ω

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

[I get cos ω = 92500 / 100000 = 0.925]
 
cytochrome said:
I've stumbled upon a system of equations that involves trig functions...

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...

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:

F=a
G=b

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:
jackmell said:
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:

F=a
G=b

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.

There is no general solution. Each problem is unique.
 
  • #10
cytochrome said:
I solved the system numerically, but I was wondering if any analytical solution exists.

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 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
freddo said:
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 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

Read more: https://www.physicsforums.com pp
 

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