# B Finding the smallest positive solution to trig equation

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1. Aug 12, 2016

### 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?

2. Aug 12, 2016

### 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.

3. Aug 12, 2016

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...

Last edited: Aug 12, 2016
4. Aug 13, 2016