To solve this problem, we can use the identity sin(2x) = 2sin(x)cos(x).
First, we need to find the value of sin(x), which we can do using the Pythagorean identity:
sin^2(x) + cos^2 (x) = 1
Since cos x = 1/3, we have:
sin^2 (x) + (1/3)^2 = 1
sin^2 (x) = 8/9
sin x = sqrt(8)/3 * sqrt(9)/sqrt(9)
= +/- (2/3)sqrt(2)
We take only positive radical since sine is positive in first and second quadrants.
Now that we know sin x, let's find sin((1/2)x):
Using the half-angle formula for sine:
(sin((1/2)x))^±²= [1-cos(x)] / ²
we get:
(sin((½)x))^±²= [¹-³]/₂=[⁴]/₆≡[+/-]√[4]/√6 ≡ [+/−][ √4 ]/[ √6 ]
Therefore,
the positive value of sin ((½)x):
[sin ((½)x)]° ≡ (+)[ √4 ]/[ √6 ]
° ≡ (∛96)/(3∛{})
≈ ..
Hence, the answer is approximately equal to ±0.177.