Find all solutions of the equation sin(2x)=-1/2

Question

From a chapter on Trigonometric Equations.

Answer 1

look on the unit circle

sin is negative in the 3rd and 4th quadrants

sin is -1/2 at 7∏/6 or 210º

sin is -1/2 at 11∏/6 or 330º

2x=210º

x=105º

2x=330

x=165º

Answer 2

sin(2x) = -1/2

 

Then, 2x = 210° + k(360°) or 330° + k(360°), k = 0, ±1, ±2, ...

 

So, x = 105° + k(180°) or 165° + k(180°)

 

If you are just looking for the solutions between 0° and 360°, plug in

k = 0 and k = 1 to get:

 

x = 105°, 165°, 285°, 345°