Question 1125018
the angles are:


-11pi/6, -7pi/6, pi/6, 5pi/6, 13pi/6


first thing you do is find the angle in the first quadrant.


sin^-1(1/2) = pi/6 radians.


multiply that by 180 / pi and it becomes 30 degrees.


this makes sense since sin(30 degrees) = 1/2.


sine is positive in the first and second quadrant.


the equivalent angle in the second quadrant is pi - pi/6 = 5pi/6 radians.


that's equivalent to 180 - 30 = 150 degrees.


5pi/6 * 180 / pi = 5/6 * 180 = 150 degrees.


that also makes sense.


these angles will repeat every 360 degrees or every 2 pi radians.


2pi radians is equivalent to 12pi / 6 radians.


pi / 6 + 12pi / 6 = 13pi / 6.


that's as far as you want to go in the positive direction.


so far your angles are pi/6, 5pi/6, 13pi/6.


in the other direction, pi/6 - 12pi / 6 = -11 pi/6.


you can't go down further than that because the limit is -16pi / 6 and subtracting another 12pi / 6 will go beyond that.


5pi / 6 - 12pi / 6 = -7pi / 6.


you can't go further than that because subtracting another 12pi / 6 will get you -17pi / 6 which is further than -16pi / 6.


your angles are therefore:


-11pi/6, -7pi/6, pio/6, 5pi/6, 13pi/6


all these angles will have 1/2 as their sine.


you can use your calculator to confirm.


alternatively, you can graph the equations to get what is shown below:


<img src = "http://theo.x10hosting.com/2018/100207.jpg" alt="$$$" >


the graph of y = sin(x) is in red.


the graph of y = 1/2 is in purple.


the intersection of the red graph and and the purple graph shows the angles where sin(x) = 1/2.


the valid interval is between -16pi/6 and 13pi/6.


that's the unshaded area of the graph.


-16pi / 6 is the same angle as -8pi / 3.


rather than simplifying all the angles, i keep the denominators the same.
it's easier to see the intervals that way.


the graphing software will, however, do the simplification where it can.