Find the values of θ in the interval [0,2π) that satisfy the equation sin2θ= -(√3)/2?

Question

you're supposed to get 4 values but i only got 2

Answer 1

sin(2x) = - sqrt(3) / 2.

From geometry of the 30-60-90 triangle:

sqrt(3) / 2 = sin(pi / 3).

A negative sine indicates quadrant 3 or 4.

As roots are required for x in [0, 2pi), you need to find all values of 2x in [0, 4pi).

Therefore:

2x = pi + pi / 3, 2pi - pi / 3, or these angles increased by 2pi.

2x = 4pi / 3, 5pi / 3, 10pi / 3, 11pi / 3

x = 2pi / 3, 5pi / 6, 5pi / 3, 11pi / 6.