solve for y in terms of x where $y= (sin(2x))^2 + (cos(x+y))^2$

Question

I tried this on Wolfram alpha but it was unable to solve it. I have looked at the graph https://www.desmos.com/calculator/rlucmlfepv and it looks possible to solve for y. can anybody solve for y.

Answer 1

This is an implicit function. Therefore, it is not possible to write $y$ solely as a function of $x$. This is the 2D plot of the function:

Another easy example to see would be: $$ x^3 + y^3 - 6xy = 0 $$

Answer 2

Consider the implicit function $$F(x,y)=y-\sin ^2(2 x)-\cos ^2(x+y)=0$$ and the simple case where $x= \frac {n\pi} 2$.

If $n$ is odd, you would need to solve $$y=\sin^2(y)$$ and if $n$ is even $$y=\cos^2(y)$$ Remember that the simple $y=\cos(y)$ does not show analytical solutions.

So, if you want solutions, fix $x$ at a given value and use a numerical method to get $y$.

For example, for $x=123.456$ (in radians); working with whole numbers, the equation to solve is $$f(y)=y-\cos ^2\left(y+\frac{15432}{125}\right)-\sin ^2\left(\frac{30864}{125}\right)$$ By inspection or graphing, you will notice that the solution is close to $1$. So, use this as $y_0$ and apply Newton method which will generate the following iterates $$\left( \begin{array}{cc} n & y_n \\ 0 & 1.000000000 \\ 1 & 1.120569912 \\ 2 & 1.177430520 \\ 3 & 1.192164485 \\ 4 & 1.193127561 \\ 5 & 1.193131547 \end{array} \right)$$