Find all real numbers that satisfy the equation sin2theta=-1/2
Let's solve the equation sin(2theta) = -1/2.
The unit circle is helpful for finding the solutions to this equation. The sine function is equal to -1/2 at an angle of -π/6 or -30 degrees and also at an angle of -5π/6 or -150 degrees. These values correspond to the points (cos(-π/6), sin(-π/6)) = (-√3/2, -1/2) and (cos(-5π/6), sin(-5π/6)) = (√3/2, -1/2) on the unit circle.
Since the sine function has a period of 2π, we can add or subtract multiples of 2π to the angles -π/6 and -5π/6 to find all the solutions.
The solutions are:
-π/6 + 2πn (where n is an integer)
-5π/6 + 2πn (where n is an integer)
These are all the real numbers that satisfy the equation sin(2theta) = -1/2.