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Tangents \( P A \) and \( P B \) are drawn to the circle \( (x-4)^{2}+(y-5)^{2}=4 \)
P from the point \( P \) on the curve \( y=\sin x \), where \( A \) and \( B \) lie on the circle. Consider the function \( y=f(x) \) represented by the locus of
W the center of the circumcircle of triangle \( P A B \). Then answer the following questions.
Which of the following is true?
(1) \( f(x)=4 \) has real roots.
(2) \( f(x)=1 \) has real roots.
(3) The range of \( y=f^{-1}(x) \) is \( \left[-\frac{\pi}{4}+2, \frac{\pi}{4}+2\right] \).
(4) None of these.
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