What is sin 2x if cos x=1/2, 0<x<180 degrees?
We notice that the values of the angle x are located in the 1st and the 2nd quadrants.
We'll apply the double angle identity:
sin 2x=sin(x+x)=sin x*cos x + cos x*sin x = 2sin x*cos x
Since the value of cos x is positive the given interval (0,pi) is stretching to (0,pi/2), because the cosine function is positive only in the first quadrant, in the second quadrant being negative.
The value for sin x is also positive in the 1st quadrant and it could be found using Pythagorean identity.
(sin x)^2 = 1-(cos x)^2
(sin x)^2 = 1 - 1/4
sin x = (sqrt 3)/2
sin 2x = 2sin x*cos x
sin 2x= 2(sqrt3/2)(1/2)
The requested value of sin 2x is: sin 2x= (sqrt3)/2.