I managed to get it in the form Rcos(θ + a) where R = 17 and a = 61.9 degrees, but I put the two expressions onto my graphical calculator and they produce different graphs! I'm not sure why this is, is it because I chose the wrong form?
I put my equation equal to 8.5 and tried solving for θ but it was negative (I got -1.9 degrees) so I added 360 to that to get 358.1 degrees. I'm now not sure why I can't find the other root.
Could someone help me? It's been happening in a few questions so I'd be very grateful.
Check your expansion of Cos(a+b), in particular the signs!
You want to be using.....
Spoiler
Edit: Also, there will be more that just one solution. Go for a general solution and get it into the form "theta = ...." before you apply any restrictions due to the required interval.
In other questions where you have to find two roots of an equation like this, what geometric reasoning would you use to find the other one?
I wouldn't, I'd use the algebra, eg.
If you had sin(x+40) = 1/2
then x+40 = 30 or 150, + n360 (where n is an integer)
so x = -10 or 110, + n360
Now if I want to restrict the interval to 0 to 360, i get the 2 solutions 110 and 350. (the first arising with n = 0 and the second with n=1; this will vary from question to question)
PS: Just to confirm what stevencarrwork said, your use of the negative angle like that is correct.