Given sin(α)=4/5 and 0<α<π/2, find the exact value of sin(α/2).
what? posting again in less than 5 minutes?
kinda antsy aren't ya?
lol steve, this website is so slow uploading i didn't think it it the first time i tried. thx though, no antsy lol
To find the exact value of sin(α/2), we can use the half angle formula for sine:
sin(α/2) = ±√((1 - cos(α))/2)
Since the given information tells us that α is in the first quadrant (0 < α < π/2), we know that cos(α) is positive. Therefore, sin(α/2) will also be positive.
Now let's find the value of cos(α):
From the given information, sin(α) = 4/5. Using the Pythagorean identity, we can find the value of cos(α):
cos(α) = √(1 - sin^2(α))
= √(1 - (4/5)^2)
= √(1 - 16/25)
= √(9/25)
= 3/5
Now, substitute the value of cos(α) into the half angle formula:
sin(α/2) = ±√((1 - cos(α))/2)
= ±√((1 - 3/5)/2)
= ±√(2/10)
= ±√(1/5)
= ±1/√5
Since α is in the first quadrant (0 < α < π/2), sin(α/2) will be positive:
sin(α/2) = 1/√5
Therefore, the exact value of sin(α/2) is 1/√5.