+129845 sin(a/2) = ± √ [ (1 - cos(a)) / 2 ] (1)
sin^2a + cos^2a = 1
(25/169) + cos^2a = 1
cos^2a = 1 - (25/169)
cos^2a = 144/169
cosa = ± 12/13
"a" will be an angle in the first or second quadrant since the sine is positive in both....so "a/2" will be in the first quadrant..and the sine and cosine will be positive here....so
a = sin-1(5/13) ≈ 22.62° or ≈ 157.38°
So.....a/2 ≈ 11.31° or a/2 ≈ 78.69°
However, "a/2" falls into the first quadrant in both cases, so the positive root in (1) is taken
So.....using (1) and assumng that a ≈ 22.62° : sin(a/2) = + √ [ (1 - (12/13)) / 2 ] ≈ 0.196
[Note that "a" falls in the first quadrant so the positive value of the cosine is used in the identity]
Verify for yourself that the sin (11.31°) ≈ 0.196
And using (1) and assuming that a ≈ 157.38° : sin(a/2) = + √ [ (1 - (-12/13)) / 2 ] ≈ 0.9806
[Note that "a" occurs in the second quadrant here, so the negative value of the cosine is used in the identity ]
And verify that sin (78.69°) ≈ 0.9806
