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:

Sin2x = Cos3x

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

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

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


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