What are the x and y components of a vector a with arrow in the xy plane if its direction is 110° counterclockwise from the positive direction of the x axis and its magnitude is 8.7 m?
Am I supposed to solve by multiplying 8.7 and 110 and cos to get x? And then multiplying 8.7 and 110 and sin to get y?
It's a fairly straight-forward solution once you draw a rough graph.
You'll see that the magnitude of the angle made to the $x$-axis can be found. Now that you have this, since length is a physical quantity, any sign convention will be the attribute of the direction it points to (i.e., $+x$-axis or $-x$-axis).
So, in this case, the angle made to the $-x$-axis, will be $70^{\circ}$. Now that you have this, treat the case normally.
The base (along the $-x$-axis) will be $8.7\cos(70^{\circ})$ and the height (along the $+y$-axis) will be $8.7\sin(70^{\circ})$.
Now that you have the magnitudes, a simple diagram will tell you that the base will be negative as it lies on the $-x$-axis and the height will be positive as it is along the $+y$-axis.
I hope this solves your problem!