Final answer:
To find tan(2x), calculate sin(x) from csc(x), use the Pythagorean identity to find cos(x), then find tan(x), and apply the double angle formula for tangent.
Step-by-step explanation:
The student is asking how to find tan(2x) given that csc(x) = 27/13 and the angle is in the first quadrant. To find tan(2x), we first need to find sin(x) and cos(x), and then use the double angle formula for tangent, tan(2x) = 2tan(x)/(1 - tan2(x)). Since csc(x) = 1/sin(x), we have sin(x) = 13/27. We can then use the Pythagorean identity to find cos(x), since sin2(x) + cos2(x) = 1. This gives us cos(x) = √(1 - (13/27)2). Once we have both sin(x) and cos(x), we can find tan(x) = sin(x)/cos(x) and then apply the double angle formula to find tan(2x).