MAT Question

https://gyazo.com/509658f8e80ec97536c78a1322acac3b

I don't really understand the question and how to approach it. I also had trouble with eliminating x or y. This question is supposed to take 5 mins, if someone could take me through their approach to this question I would appreciate it.

If you convert those simultaneous equations into matrix form then you have a rotation matrix for $\theta$ anticlockwise about the origin. There are two ways to view this, the simplest is that the determinant is always 1 so there exists an inverse matrix for all values of theta (though since the trig functions have period $2 \pi$ this is equivalent to $0 \leq \theta < 2 \pi$). So for all theta the equations have a unique solution.

Alternatively rotations are bijective. There must be a point mapping onto (2, 1) whatever $\theta$ is.

Disclaimer: No idea if this is right.

I haven't read through this properly but it is not really suitable of the MAT. The best way to do this question is let sin(theta)=s and cos(theta)=c and treat like a usual simultaneous equation

I tried doing that however I don't really understand what the 4 choices even mean.

I got something like 2c - s = x and something similar when eliminating y.

I don't get how this answers helps me towards making a choice.

for every value of theta you get a unique value of x and y that solves the equations

Would it ever make sense that there could not be a value of theta which was invalid.

If the equation you formed was something like x = c/s could you say that there would be 1 value of theta ( theta = 0 ) in the interval which would not be valid?