# 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.

Thanks in advance! - Shaun

• Yes, that is correct. f(x)=g(x) is always the set of all the solutions (x values of points of intersection). – MrYouMath Sep 17 '15 at 12:10
• Yes!, The answer is where the two graphs are intersecting – SMA.D Sep 17 '15 at 12:10

## 1 Answer

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 $$f(x)=g(x)$$.