non linear implicit equation

Good morning, I'm new in using Matlab and I would like to know if it is possible to solve the following implicit equation

sin(x)-y*sin(acos((3+sin(x))/y)) = 0

where: x = linspace(0,2*pi,50)

In particular I would like to explicit y and after plot y against x.

Thank you so much!

Answer 1

[Alan Stevens]

Surprisingly, this has an explicit solution for y (I didn't use Matlab to get it as I don't have the symbolic toolbox to hand) . See the following code:

x = linspace(0,2*pi,50);
y = zeros(1,numel(x));
for i = 1:numel(x)
    y(i) = yfn(x(i));
end
subplot(2,1,1)
plot(x,y),grid
xlabel('x'),ylabel('y')
subplot(2,1,2)
plot(x,sin(x),'b',x,y.*sin(acos((3+sin(x))./y)),'r--'),grid
xlabel('x'),ylabel('check functions')
legend('sin(x)','y.*sin(acos((3+sin(x))./y))')

function y = yfn(x)
            
             y = sqrt(6*sin(x)-cos(2*x)+10);
             if mod(x,2*pi)>pi
                y = -y;
             end
             
end

This results in

[Gregory Cottone]

Thank you Alan,

I don't understand how did you find this function y = sqrt(6*sin(x)-cos(2*x)+10).

Could you kindly explain me?

[Alan Stevens]

As follows:

I've just taken the positive sign for y above, but you need the negative sign between pi and 2pi.

Hope that helps.

[Gregory Cottone]

Thank you very much!!

One more last question: if I would find a numerical method as for example fsolve function, what I have to do to set the problem?

[Alan Stevens]

fsolve is part of the Optimisation toolbox, which I don't have, so I'm not much help to you on that. I suggest you either type doc fsolve, or ask as a separate question in the forum (or both!).