Finding the smallest positive solution to trig equation

[Mr Davis 97]

I have the equation $\sin 3x = \cos 7x$, and, in degrees, I have to find the smallest positive solution.

Immediately, we can see that sin and cos are equal if their arguments are complements, so $3x + 7x = 90$, which means that $x = 9$.

I know that that is a correct solution, but how do I show that it is, in fact, the smallest positive solution?

 

[Isaac0427]

I would do it this way:
sin(x)=cos(x) if x=45 degrees. Of course, you knew that and could then get the correct solution. However, you are missing one part -- x=45+180n degrees where n is an integer. If you are looking for the smallest possible x, n would have to be zero.

 

[Charles Link]

An interesting problem. Since the angles are both in the first quadrant, I think you clearly have the smallest x. To prove it might take a little work, but it would take more effort to find the second smallest or 3rd smallest. (Solving the equivalent $\sin(3x)=\sin(90-7x)$, trigonometric identities allow 360 degrees to be input into either side and you have essentially the same equation ) e.g. If $3x=360+(90-7x)$ so that $10x=450$ then x=$45$. I found another solution, but is it the second smallest? editing... You can also add 720 or other multiples of 360 to either side, or you can do $180-\theta$ to either term inside the sine function without changing the equation...

 

[Charles Link]

A follow-on to post #3: Overlaying a graph of y =sin(3x) and a graph of y=cos(7x) is perhaps the quickest way to see the (approximate) solutions of this problem. It will show you that your x=9 is the smallest x.