# Help making x the subject of a trigometric equation containing $x$ and $\sin(x)\cos(x)$

I am trying to rearrange the following formula to make x the subject. $\frac{x}{180^\circ}\cdot \pi - \sin(x)\cos(x) = \frac{\pi}{y}$. I appreciate that I could use the double angle identity in reverse, but this still isn't getting me anywhere. This is not for school, I am playing with an idea and the math is getting a bit above me. Thank you

• Welcome to MSE. Please use MathJax to format your questions. – user507623 Dec 26 '17 at 9:55
• you can convert $\sin(x),\cos(x)$ into $\tan(\frac{x}{2})$ – Dr. Sonnhard Graubner Dec 26 '17 at 9:59
• The simplest way is, perhaps, to put $\;\sin x\cos x=\frac12\sin2x\;$ ...Still, you get an equation in which you won't be able to "separate" $\;x\;$ by means of elementary functions. – DonAntonio Dec 26 '17 at 10:01

Do not get discouraged. It is not always possible. Actually, it is very rarely possible. Those school examples where it is possible are just carefully chosen precisely so that it is possible; for a random formula with $x$ and $y$ it would be a surprise to be able to extract one of the variables as a function of the other.
A historical note: look up the problem of solving equations in radicals. You may know that, if you have $y=ax^2+bx+c$, you can solve it for $x$ as a quadratic equation and get the solution expressed using the square root... People have solved the same for a cubic and quartic expression in $x$, and then got stuck for centuries trying to solve the 5th degree equation (in effect solve $y=ax^5+bx^4+cx^3+dx^2+ex+f$ for $x$). In the 19th century, based on the work of Galois, Abel, Gauss and others, it was proven that it is not possible (at least not using roots), but not before this work created foundations for completely new (at the time) branches of algebra such as group theory and field theory.