Whilde defining sin(theta), where did you get y and r from? Are you using a right angled triangle with an angle theta to determine the trigonometric identities? Please state clearly.

similarly, csc(theta) = 1/sin(theta)

sec(theta)= 1/cos(theta)

cot(theta) = 1/tan(theta)

arc sin is the same as csc!

arcsin(theta)= (inverse of sin(theta)) = 1/sin(theta) = csc(theta)