Check whether the centre of the circle is within the rectangle expanded by the radius in all directions. If not, the answer is "no". Else, check whether the centre of the circle is within the rectangle. If so, the answer is "yes". Else for each line, check whether the centre of the circle is less than the radius away from it, and if so, calculate the angles at which the line intersects the circle. If any of these angles is in the angle range of the sector, the answer is "yes". Else the answer is "no".
This avoids unnecessary line/circle intersections when the answer can easily be determined without them. Note that it relies on the fact that the border of the rectangle can't intersect the border of the sector without also intersecting its circular part.