Question

Given sin(x)=7/9 pi/2<α<pi. Find the exact value of sin(x/2). (With no decimals)

Answer 1

Use Half-angle formula:

`sin ((x)/(2)) = \sqrt{(1-cos x)/(2)}`

We need cos(x), which can be found using pythagorean identity:

`sin^2 + cos^2 = 1`
Note that the angle is in 2nd quadrant indicating that cos(x) is negative.

`cos(x) = -√(1 - (7/9)^2) = -\sqrt{(81-49)/(81)} = -(4 √(2))/(9)`

Substitute this value into the half-angle formula:

`sin((x)/(2)) = \sqrt{(1 - (-4√(2)/9))/(2)} = \sqrt{(9+4 √(2))/(18)}`