+129928 We want to know where the sin = 1/2......
This occurs at 30° + n* 360° and at 150° + n*360° where n is an integer
Here is a graph which confirms this........https://www.desmos.com/calculator/03z7sxndpl
+1904 \(sin(x)=1/2\)
To isolate the x, take the sine inverse of both sides to get your first answer
\(x=sin^-1(1/2)\)
To take the sine inverse of 1/2, you have to find a angle bweteen \(-\pi/2\) and \(\pi/2\) (-90° and 90°)
To do that you need to refere to the unit circle. Go to the following website to look at the unit circle: http://www.thattutorguy.com/unit-circle-pdf/
When you look at the unit circle, look for which angle between \(-\pi/2\) and \(\pi/2\) (-90° and 90°) whose y value (sine) is equal to 1/2.
When you do that, you will find that \(x=\pi/6\) or 30°.
You will notice on the unit circle there is another angle not between \(-\pi/2\) and \(\pi/2\) (-90° and 90°) whose y value (sine) is equal to 1/2.
If you look at qadrant II you will see that the angle whose y value (sine) is equal to 1/2 is \(5\pi/6\) or 150°. You can then say that \(x=5\pi/6\) or 150° which would be answer two.
You think those are the only two answers. No way. In fact, there are infinate answers. To find all of the answers, you take the two answers you got from the unit circle and if you add \(2\pi\) or 360° you, will get two more answers. If you add \(2\pi\) or 360° to those answers, you will get two more answers and you and keep doing this forever to get infinite answers.
Of course it is not practical to list every answer which would take forever so instead to reprecent every aswer, form two equations that would represent every answer.
Those equations are: \(x=\pi/6+2\pi*n\) (\(x=30°+360°*n\)) and \(x=5\pi/6+2\pi*n\) (\(x=150°+360°*n\)); n is an intenger.
