If \(\theta=\frac{m\pi}{18}\) lies on the negative (x)-axis and \(0<\theta<4\pi\) then what is the sum of possible values of (m)?

Correct answer: (72)

A point lies on the negative x-axis when its angle is an odd multiple of \\(\\pi\\): \\(\\pi,3\\pi,5\\pi,\\ldots\\). The condition here is \\(0<\\theta<4\\pi\\), so only \\(\\theta=\\pi\\) and \\(\\theta=3\\pi\\) are included. Substituting \\(\\theta=m\\pi/18\\), we get \\(m\\pi/18=\\pi\\), giving \\(m=18\\), and \\(m\\pi/18=3\\pi\\), giving \\(m=54\\). Their sum is \\(18+54=72\\), so option C is correct.

The strict inequalities matter: zero is excluded, and \\(4\\pi\\) itself is also excluded. Although \\(5\\pi\\) is again on the negative x-axis, it is outside the stated interval. Angles differing by \\(2\\pi\\) point in the same direction, which explains the two permitted positions, \\(\\pi\\) and \\(3\\pi\\), within this interval. Thus the supplied answer C is correct and the explanation identifies all possible values without adding an invalid endpoint.