So my buddy and I (both HS Math teachers) have been messing around with a question about finding all solutions of a "co-function" equation like the one above. The typical HS questions asks students to find the solution in Q1 by solving $2x + 3x = 90$.
He was thinking about how to get the other solutions and we were curious to see what other people do for a problem like this.
We've noticed that the solutions come in regularly spaced intervals from two separate starting points.
Namely, $\frac{\pi}{10} \pm \frac{4k\pi}{10}$, AND $-\frac{\pi}{2} \pm 2k\pi$ {$k$ natural} for the above equation.
We've been able to generalize this to any such co-function equation of the form $\sin(ax)=\cos(bx)$ {$a,b$ Integers} , and are curious about whether other people have stumbled upon this as well.
We've also discovered a very interesting geometric interpretation of the periodic solutions and where they occur.
We'd love to hear how you approach this type of problem (algebraically, not graphically) and whether you've encountered these ideas as well.
Best Answer
$\sin(2x) = \cos(3x) \Rightarrow \cos(\frac{\pi}{2}-2x) = \cos (3x) \Rightarrow \dfrac{\pi}{2} - 2x = \pm 3x + 2k\pi, k \in \mathbb{Z}$. Can you take it from here?