Given sin(α)=4/5 and 0<α<π/2, find the exact value of sin(α/2).
Draw your triangle. It is just a 3-4-5 triangle, so cosα is easy to read off it.
Now use the half-angle formula:
sin(α/2) = √((1-cosα)/2)
Ty!
To find the exact value of sin(α/2), we can use the half-angle formula for sine. The formula is:
sin(α/2) = ±√[(1 - cos(α))/2]
To determine the sign, we need to consider the quadrant in which α lies. We know 0 < α < π/2, which means α is in the first quadrant. In the first quadrant, both sine and cosine are positive, so the sign will be positive.
Now, let's find the value of cos(α). We can use the identity cos^2(α) + sin^2(α) = 1, substituting the given value of sin(α) = 4/5:
cos^2(α) + (4/5)^2 = 1
cos^2(α) + 16/25 = 1
cos^2(α) = 1 - 16/25
cos^2(α) = 25/25 - 16/25
cos^2(α) = 9/25
Taking the square root of both sides, we get:
cos(α) = ±√(9/25)
cos(α) = ±3/5
Since α is in the first quadrant, cos(α) is positive, so we have:
cos(α) = 3/5
Now we can substitute this value back into the half-angle formula:
sin(α/2) = √[(1 - cos(α))/2]
sin(α/2) = √[(1 - 3/5)/2]
sin(α/2) = √[(2/5)/2]
sin(α/2) = √[1/5]
sin(α/2) = 1/√5
So, the exact value of sin(α/2) is 1/√5 or √5/5.