# Solving a nasty set of equations

• Mr.Miyagi
In summary, the conversation discusses a set of equations involving x, y, r, and theta that were distilled from a geometrical problem involving the tangent of two circles. The goal is to find an analytical solution for the variables, but the conversation highlights difficulties in finding a solution and suggests using trigonometric identities. The geometry of the problem is explained through a drawing, and it is mentioned that the line BC has a length of (3π-1)r. The conversation ends with the desire to find the length of line DG and the angle alpha.
Mr.Miyagi
Suppose I have the following set of equations
$$x=3\pi r-r-2\sin\theta$$
$$y=r-2\cos\theta$$
$$\tan\theta=y/x$$
with r as a constant

How would I go about finding out if an analytical solution exists? There must be a solution, as I distilled the equations from a fairly straightforward geometrical problem that I conjured up.
I just can't seem to find it.

The geometrical problem is basically finding the length of the inner tangent of two circles, between the points where the tangent and the circles touch plus parts of the circles. It's length would be $$\sqrt{x^2+y^2}$$ and the parts of the circle have a length $$2\theta r$$. I hope this makes sense.

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Hi Mr.Miyag!

(have a theta: θ )

Rewrite sinθ and cosθ in terms of tanθ,

then convert them to an expression in y/x, and solve.

Hi:)

I still don't see how I can solve for all three variables.
$$\sin\theta=\tan\theta\cos\theta=\cos\theta\frac{y}{x}$$
$$\cos\theta=\tan\theta\frac{\cos^2\theta}{\sin\theta}=\tan\theta\cot\theta\cos\theta=\cot\theta\cos\theta\frac{y}{x}$$
then
$$x=(3\pi-1)r-2\cos\theta\frac{y}{x}$$
$$y=r- 2\cot\theta\cos\theta\frac{y}{x}$$
I don't see a substitution that simplifies things.

Hi Mr.Miyagi!

(what happened to that θ i gave you? )
Mr.Miyagi said:
… I don't see a substitution that simplifies things.

You need to learn your trigonometric identities …

in particular, sec2θ = tan2θ + 1

(What's up with the theta?)

Ah, I overlooked the identity. I can express x as a function of y and vice versa, with the trig identity now. But it doesn't seem to help me in finding θ as a function of r.(I shouldn't have said r is a constant. It is just a variable).
I feel I'm overlooking something really obvious, but I can't seem to figure it out.

I don't quite understand the geometry you are trying to explain. How are these circles defined again? Are they concentric.

Also, if your x,y,r,and theta are both the coordinates, you could use any of the equations that do {x,y}-->{r,theta} and visa versa.

For example,

x = r Cos[theta]
y = r Sin[theta]

I apologize for the poor choice of variable names. I've made a drawing to explain what I was talking about. It's in the attachment.
The line GH would be our x, DH would be our y, the radius of the circles is our r and the angles denoted apha are our theta.
It is also known that the line BC has length $$(3\pi-1)r$$
Now I'd like to know the length of the line DG and the angle of alpha.

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## 1. How do I approach solving a set of equations?

The first step in solving a set of equations is to make sure they are written in standard form, with all variables on one side and constants on the other. Then, you can use algebraic techniques such as substitution, elimination, or graphing to find the solution.

## 2. What should I do if I get stuck on a difficult equation?

If you are having trouble solving a particular equation, try breaking it down into smaller, simpler steps. You can also ask for help from a classmate, teacher, or online resources.

## 3. Can I use a calculator to solve equations?

In most cases, yes, you can use a calculator to solve equations. However, it is important to understand the underlying concepts and techniques so you can check your answer and ensure it is correct.

## 4. How can I check if my solution is correct?

You can check your solution by substituting the values you found for the variables back into the original equations. If the equations are satisfied, then your solution is correct.

## 5. Are there any tips for solving equations more efficiently?

Yes, there are some tips that can help you solve equations more efficiently. These include being organized and writing out each step clearly, using shortcuts such as factoring, and practicing regularly to improve your skills.

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