What is the value of cos x if sin2x=2/3? x is in interval (pi/2,pi)
To determine cos x, we'll have to apply the half angle identity:
cos x = +/- sqrt [ (1 + cos 2x) / 2 ]
We know, from enunciation, that:
pi < x < pi / 2
We'll multiply by 2 the inequality:
2pi = 0 <2x < pi
From the above inequality, the angle 2x covers the 1st and the 2nd quadrants and the value of cos x is positive in the 1st quadrant and negative in the 2nd quadrant.
Since sin 2x = 1/4, we'll apply the trigonometric identity
(sin 2x)^2 + (cos 2x)^2 = 1 to determine cos 2x,
cos 2x = +/-sqrt(1 - sin 2x)
cos 2x = +/- sqrt(1 - 4/9)
cos 2x = +/- sqrt(5) / 3
We'll substitute cos 2x by its value in the formula for cos x and we'll keep only the negative values for cos x, since x is in the 2nd quadrant, where cosine function is negative.
cos x = - sqrt [ (1 + cos 2x) / 2 ]
cos x = - sqrt [(3+sqrt5)/6]
cos x = - sqrt [(3-sqrt5)/6]
The requested values of cos x are: cos x = - sqrt [(3+sqrt5)/6] and cos x = - sqrt [(3-sqrt5)/6].