How many solutions do these simultaneous trig equations have?

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The simultaneous equations in \(x\),\(y\), \[(\cos\theta)x - (\sin\theta)y = 2\] \[(\sin\theta)x + (\cos\theta)y = 1\] are solvable

1. for all values of \(\theta\) in the range \(0 \leq \theta < 2\pi\);
2. except for one value of \(\theta\) in the range \(0 \leq \theta < 2\pi\);
3. except for two values of \(\theta\) in the range \(0 \leq \theta < 2\pi\);
4. except for three values of \(\theta\) in the range \(0 \leq \theta < 2\pi\).

In this applet, the blue line is \(x\sin \theta + y\cos \theta = 1\), and the red line is \(x\cos \theta - y\sin \theta= 2\).

How will each of the lines change as we change \(\theta\)?

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The intersection point, \(A\), represents the solution to the simultaneous equations. For which values of \(\theta\) is the point \(A\) uniquely defined?