Final answer:
The angle θ is located in quadrant I, as it is the only quadrant where both the sine (csc θ is positive) and tangent functions are positive.
Step-by-step explanation:
The problem asks us to determine the quadrant in which an angle θ is located given that the cosecant of the angle (csc θ) is 13/5 and the tangent of the angle (tan θ) is positive. Recalling that the cosecant function is the reciprocal of the sine function (csc θ = 1/sin θ), we note that the sine of angle θ must be positive because 13/5 is a positive value.
This information narrows our search to quadrants I and II, where the sine function is positive. Since we are also given that the tangent function is positive and the tangent function is positive only in quadrants I and III, the only quadrant that satisfies both conditions is quadrant I. Therefore, angle θ is located in quadrant I.