I'm busy working through a section on trigonometric equations in my textbook and not having any problem solving them algebraically, but I'm not entirely sure I have a complete understanding of what is really going on and what the equations are describing. I've made an attempt below at what I think is happening.
For instance, $$\cos(3x) = \sin(2x)$$
Algebraically, simple enough to solve:
$$\sin(2x) = \sin(90^{\circ}-3x)$$ \begin{align} 2x &= 90^{\circ} - 3x + k\cdot360 ^{\circ} \\ 5x &= 90^{\circ} + k\cdot360^{\circ} \\ x &= 18^{\circ} + k\cdot72^{\circ} \end{align} \begin{align} 2x &= 180^{\circ} - (90^{\circ}-3x) + k\cdot360^{\circ} \\ -x &= 90^{\circ} + k\cdot360^{\circ}\\ x &= -90^{\circ} + k\cdot360^{\circ} \end{align}
This is how I would graph that equation using a graphing program:
Would I be correct in saying that graphically $\sin(2x) = \cos(3x)$ describes where the two graphs are intersecting? I'd appreciate any elaboration or correction on this topic.
Thanks in advance! - Shaun