Max. and Min.

Consider g(a) = f(-a) = \frac{1-\sin a}{3-\cos a}

This may be interpreted as the slope of the line joining the point (3,1) to the unit circle any point on which is given by (\cos a, \sin a)

Now the question translates to the maximum and minimum of slopes of lines joining (3,1) to any point on the unit circle. The corresponding lines are obviously the tangents from (3,1) to the unit circle.

One of them is easy to see, which is a line parallel to x-axis and passing through (3,1) which has slope zero. This is the minimum.

Again using geometry the other slope equals 2 \tan^{-1}\frac{1}{3} = \frac{3}{4}