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Question: Consider the diagram below. How can one express the length of $z$ in terms of $a$, $b$, and $c$ without including $x$ in the expression? If it is impossible to express $z$ in terms of $a$, $b$, and $c$ alone, please answer with an explanation of why.

This is the diagram.

More Information: $ax$ and $bx$ are angles, where $x$ is some constant and $a$ and $b$ are each being multiplied by it. In other words, the two angles are in ratio $a$ to $b$. You can assume that $a$ is greater than $b$ and both $ax$ and $bx$ are less than 90 degrees. $c$ is the length of the line segment along the diagram's base.

Motivation: I'm asking this question because I feel that the constraints on the diagram are sufficient to bind $z$ to a single value for given $a$, $b$, and $c$. That is, I suspect knowing the values of $a$, $b$, and $c$ (and knowing the other lengths I have specified) one should be able to determine $z$ with certainty.

Thus, I suspect it is likely $z$ can be expressed in terms of only $a$, $b$, and $c$. Unfortunately, all my attempts to derive an expression for $z$ have ended up with me including $x$ in my expression or just churning out tautologies. Clearly help is needed. Thanks in advance!

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    Perhaps it is notable that $c = \sin(ax) + \sin(bx)$. – 2017-01-12
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    Also, $z = 2 \sin(\frac{(a + b)x}{2})$ – 2017-01-12
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    Also, $$ z^2 = c^2 + (\cos(bx) - \cos(ax))^2 $$ – 2017-01-12
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    Definitely all important observations! Also, there are a number of similar triangles (3 it looks like) scattered throughout the bottom half of the diagram. I've tried to mess with those, but no luck. – 2017-01-12

2 Answers 2

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I asked Wolfram Alpha to find $\cos\left(\frac25x\right) - \cos(x) = 1$ (the square root of $z^2 - c^2$) where $\sin\left(\frac25x\right) + \sin(x) = 1$. It told me the exact forms are roots of a ninth-degree polynomial.

Setting $b = \frac23$ instead of $\frac25$ resulted in a fifth-degree polynomial, while $b = \frac27$ resulted in one of the thirteenth degree.

I don't think you're ever going to find a useful closed formula that will work for $a$ and $b$ in general. I would recommend a numeric approach: find an approximate solution for $x$ in $\sin(ax) + \sin(bx) = c$ to any degree of accuracy you want. This should be easy enough to do (for example by the bisection method), because $\sin(ax) + \sin(bx)$ is a strictly increasing function from $x=0$ to $x=\frac\pi{2a}$. Moreover, it works for arbitrary values of $a$ and $b$, not just ones such that $\frac ba$ is a ratio of integers with a small denominator.

Once you have a value of $x$, use it to compute $$z = \sqrt{c^2 + (\cos(bx) - \cos(ax))^2}.$$

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Let's call $d$ the height of the rectangle, then:

$$d=(d+e)-e=\cos (bx)-\cos (ax) \quad (1)$$

we also know that

$$p+q=\sin (ax)+ \sin (bx)=c \quad (2)$$

then

$$z^2=d^2+c^2=2(1-\cos [x(a+b)])$$

From $(2)$ we can hope to find $x$ once $a,b,c$ are properly given. Once we find $x$ as a function of $a,b,c$ then $z$ doesn't depend on $x$.

The big problem is solve $(2)$. We can prove if there is a solution once we know $a,b,c$ but find a close formula I don't know if it is possible.

For example: how to find the solution for:

$$\sin(11x)+\sin(61x)=0.1234$$

That will be (roughly speaking) a $61$ degree polynomial and we should find a roots for it.

What I can say is the following:

If we know $a,b,c$ then we can say if $(2)$ has a solution. If there is a solution then $z$ doesn't depends on $x$. Find a close formula is something that I can't assure. Probably you will need a numerial method to find an approximation.