What is the value of cos x if sin2x=2/3? x is in interval (pi/2,pi)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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].

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