This is not actually homework per se - it's not worth marks directly - I'm just studying the UTexas page on driven damped harmonic motion and trying to work through it.

https://farside.ph.utexas.edu/teaching/315/Waveshtml/node13.html

But I've hit this roadblock and am hoping someone(s) can point me in the right direction - either with some simple error I'm making or by telling me I'm using the wrong method in the first place.

The part I'm stuck at (and have been stuck at literally for hours) is this:

So since they tell me the two expressions can be combined, I'm trying to combine them.

I got 110, like so:

But can't for the life of me work my way through to 109.

My first attempt was the following, using only 107, which has everything in its place except for those pesky trig functions:

So I figured the way to get rid of a sin and a cos is do some squaring, so they'll hopefully sum and drop out as a 1. So I squared 107 and 108 and set them equal to each other (since they both equal 0). As you can see here below though, I still haven't been able to get it to where it needs to be, because the LHS is not a sum, it's a subtraction:

If I had a + instead of a - on the left I would be much happier, but I can't see how and now I've been trying to nut this out for so long I'm doubting I've even gone down the right road.

Any advice would be much appreciated.

 

Like this ....

 

Hi,

Real quick, i didnt go over this thoroughly but there may be a problem with the sign of the result after solving for x0 because there may be an absolute value in there somewhere. So that might mean the result could be plus or minus. I could be wrong though, so maybe check it over if you wish.

Once you find the angle you can substitute it into the first equation then solve for x0 because cos(tan) and sin(tan) simplify (or similar trig reductions).

 

Like this ....

Thanks t_n_k, that's great. I was obviously down the wrong track after all.