cos x = -1/3, x in quadrant III. Find the value of sin(x/2), cos(x/2), tan(x/2)

cos x = -1/3, π < x < 3π/2
i.e. x lies in 3rd quadrant
Using 1 - cos x = 2sin²(x/2) ⇒ sin(x/2) = ±√(1 - cos x)/2
We get, sin(x/2) = ±√(1 - (-1/3))/2 = ±√(4/6)
As π < x < 3π/2 ⇒ π/2 < x/2 < 3π/4 and sin is positive in 2nd quadrant
∴ sin(x/2) = √(2/3)
Using 1 + cos x = 2cos²(x/2) ⇒ cos(x/2) = ±√(1 + cos x)/2
we get, cos(x/2) = ±√(1 + (-1/3))/2 = ±√(2/6)
As π < x < 3π/2 ⇒ π/2 < x/2 < 3π/4 and cos is negative in 2nd quadrant
∴ cos(x/2) = -√(1/3)
Using cos x = (1 - tan²(x/2))/(1 + tan²(x/2)) ⇒ tan(x/2) = ±√((1 - cos x)/(1 + cos x))
We get tan(x/2) = ±√((1 - (-1/3))/(1 + (-1/3))) = ±√(2)
As π < x < 3π/2 ⇒ π/2 < x/2 < 3π/4 and tan is negative in 2nd quadrant
∴ tan(x/2) = -√2