Verified answer

 Quadrant IV is between 270º and 360º

  tan[t] = -1 ⁄ (0.9)  ...  cot[t] = 1 ⁄ tan[t]

    t = tan‾¹[-1 ⁄ (0.9)] = -48º  or  -48 + 360 = 312º

    t = 312º = 312 • (π ⁄ 180) = 5.445 radians

 The rest can be done with a calculator:

  Working in radians ...

   sin[5.445] = -0.743

   cos[5.445] = 0.669

   csc[5.445] = 1 ⁄ sin[5.445] = -1.345

   sec[5.445] = 1 ⁄ cos[5.445] = 1.5

   sin[t + 2π] = sin[t] = -0.743 ... repeats every 2π radians