# Approaching Trigonometric Equations graphically.

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.

Just to summarize what was said in the comments, you're exactly right. Whenever you graph two functions $$f(x)$$ and $$g(x)$$, the $$x$$-values of the points at which the two functions intersect will be the same values of $$x$$ that satisfy the equation $$f(x)=g(x)$$.