[1]" "arcsin[cos((10pi)/9)]
Convert cos((10pi)/9) to sin using the co-function identity.
[2]" "=arcsin[sin(pi/2-(10pi)/9)]
Simplify.
[3]" "=arcsin[sin((9pi)/18-(20pi)/18)]
[4]" "=arcsin[sin((-11pi)/18)]
The restricted domain for sine is [-pi/2,pi/2] since this involves an arcsin. We must look for an angle in this restricted domain whose sine has an equal value to sin((-11pi)/18). That angle is (-7pi)/18 because it has the same reference angle and sign as (-11pi)/18.
[5]" "=arcsin[sin((-7pi)/18)]
Arcsin and sin will cancel because of the definition of inverse functions.
[6]" "color(blue)(=(-7pi)/18)
