solve

Question

sin (2Θ/3) = -0.500, 0 degrees <= Θ => 360 degrees?

Answer 1

The trig equation is sin(2θ/3) = - 0.500

sin(2θ/3) = - 0.5

sin(2θ/3) = - 1/2

Let 2θ/3 = t ⇒ sin(t) = -1/2

The function sinx is negative in third and fourth quadrant.

In third quadrant -1/2 = sin(π + π/6) = sin(7π/6)

In fourth quadrant -1/2 = sin(2π - π/6) = sin(11π/6)

So the solutions are t = 7π/6 and 11π/6.

Finally, add multiples of 2π to each of these solutions to get the general form.

t = 7π/6 + 2nπ  and t = 11π/6+ 2nπ where n is integer.

2θ/3 = 7π/6 + 2nπ and 2θ/3 = 11π/6 + 2nπ

θ = (7π/6 + 2nπ)3/2 and θ = (11π/6 + 2nπ)3/2

The solutions are θ = (7π/4) + 3nπ and θ = (11π/4) + 3nπ.

The solution is θ = 7π/4 in [0, 2π).