Question

Sin(θ) = − 9 41 where 3π 2 < θ < 2π

Answer 1

sin theta = -9/41, where theta is in Q IV; find the angle theta:

Here the opp side is -9, the hyp. is 41.  The adj side is found by applying the Pyth. Thm.:

(-9)^2 + x^2 = 41^2

81      + x^2  = 1681.  Then x^2 = 1600, and the adj. side (x) is +40.  We know it's +40 because theta is in Q IV.

The angle theta is  arcsin (-9/41).  Using a calculator, 

arcsin (-9/41) = -0.22 radians (which is correct because theta is in Q IV, just as -0.22 rad is in Q IV.    You could express theta as a positive angle by subtracting 0.22 rad from 2pi rad:  1.78 rad.  You could, of course, change these angles in radians into angles in degrees.