The graph above illustrates a unit circle and the graph of \(y=\sec(x)\). There are two sliding points. On the figure of the unit circle, you can slide the red point up and down, representing a value \(s\) giving the target secant for an angle. On the figure for the graph of secant, you can slide the blue point, representing an angle to visualize on the unit circle.
Because the secant and the reciprocal of the cosine are related, a line showing the target cosine value is also illustrated.
When you adjust the target \(s\) value, the terminal ray of an angle \(\theta\) such that \(\sec(\theta) = s\) with \(\theta \in [0,\frac{\pi}{2}) \cup (\frac{\pi}{2},\pi]\) is also illustrated in red (if possible). This angle is the definition of \(\theta = \sec^{-1}(s)\). You can see that there are other angles that satisfy \(\sec(\theta)=s\) (infinitely many of them, in fact).
What is the relationship between all of these angles?
Adjust the angle \(\theta\) using the slider on the right so that you can find angles satisfying \(\sec(\theta)=s\). There are two families of solutions. How are they related? Can you describe both families as a single set?