Question

Find all the soultions of the equation in the interval (0,2pi)

sin 2x = -sqrt3 /2

sin -sqrt3 /2 is 4pi/3 and 5pi/3 in the unit circle

2x= 4pi/3 + 2npi 2x=5pi/3 + 2npi

I do not know what to do at this point.

Answer 1

30, 60, 90 triangle
1, sqrt 3,sides and 2 hypotenuse

sin A = + (sqrt 3)/2 if A = pi/3
so
if A = pi + pi/3 which is 4 pi/3
or
if A = 2 pi - pi/3 which is 5 pi/3

then sin A = -sqrt(3) /2 {AGREE}
But A = 2x
so
x = 2 pi/3 or 5 pi/6

Now if they mean x can be as large as 2 pi we better look for bigger angles for A
Like A = 2 pi + 4 pi/3 = 10 pi/3
then x = 5 pi/3
and
Like A = 2 pi + 5 pi/3 = 11 pi/3
then x = 11 pi/6
which is still less than 2 pi (barely
so
I get four values for x total