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:
$sin (\frac{x}{2}) = \sqrt{\frac{1-cos x}{2}}$

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.
$cos(x) = -\sqrt{1 - (7/9)^2} = -\sqrt{\frac{81-49}{81}} = -\frac{4 \sqrt{2}}{9}$

Substitute this value into the half-angle formula:
$sin(\frac{x}{2}) = \sqrt{\frac{1 - (-4\sqrt{2}/9)}{2}} = \sqrt{\frac{9+4 \sqrt{2}}{18}}$