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I need to solve this simple transcendental equation, but I don't exactly know how.

$cos(x - 2y) + 8x + 4y = 0$

I mean, I think I've never learnt any method to try solving an equation like this. The only way I can think of is the bisection method, but it would be quite long. Is there something I'm missing?

I'm sorry, I'm quite new to exercises of this kind.

Lennard.
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  • Just to clarify, you're particularly interested in numerical solutions, right? – Matti P. Jul 21 '20 at 07:37
  • Yes! I need numerical solutions because these are the stationary points in which the first derivative becomes zero. – Lennard. Jul 21 '20 at 07:39
  • The first thing that we see is that there are of course infinitely many solutions. Then, because cosine is always between -1 and 1, we see that in the grand scheme of things, the solutions are not too far away from $$ 8x+4y=0 $$ So for any $x$, the a good candidate for the first guess for a solution is $y\approx -2x$ – Matti P. Jul 21 '20 at 07:43

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Let $x-2y=z.$ Then $0=\cos z+8x+4y=\cos z+8z+20y,$ so $$y=-(8z+\cos z)/20\;..... (1)$$ and $$x=z+2y=z-(8z+\cos z)/20\;..... (2)$$ Putting any $z$ into $(1)$ and $(2)$ will give a solution for $x,y.$

Don't try to find a simple formula for $y$ in terms of $x,$ or vice-versa. There isn't one.

DanielWainfleet
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