A Level Maths Trig Question

Maybe an observation (you've probably spotted it), is that the LHS of the equation in (b) is "practically the same" as in (a).
So the thinking here is to set theta to be something appropriate to "make (b) happen".
What is the "appropriate something"? Well, if you compare the two LHS's, it should be clear what it is.

Then it's routine, hopefully.

How do you do a?

I'm aware that you can equate 7sin(theta + 51.8) to 3, (and can equate the 5cos bit to the bit in part a) but when I do this and try to solve for theta, I get a negative angle. What would I then do with the negative angle? Would I need to use a CAST diagram to find positive equivalent angles?

Ah, okay.
If theta satisfies sin(theta)=blah, then so does 2npi+(pi-theta) and 2npi+theta (or 180 degrees for pi, if you don't know radians).
So really "just try" from here on.

As an example, if 30 degrees is a solution to an equation in sine, then so does 150, 150+360, 150+720... and 30+360, 30+720,... In fact 30-360 etc. also works.

I tend to remember this by thinking sine means the y-coordinate of the point on the unit circle, and these two "transformations" don't affect the y-coordinate (pi-theta means reflection along y-axis; 2npi+theta means going around the circle n times back where the point originally is.) That said, you might have other ways of remembering it, be it CAST, rote memory, graphs, whatever you choice is.