What is the value of sin (a/2) if sin a=0.25? 90<a<180
Since we have to determine the sine of the half angle, we'll apply the formula:
sin (a/2) = +/- sqrt [(1 - cos a)/2]
We know, from enunciation, that:
90 < a < 180
We'll divide by 2 the inequality:
90/2 < a/ 2 < 180 /2
45 < a/2 < 90
From the above inequality, the angle a/2 is in the 1st quadrant and the value of sin (a/2) is positive.
Since we need the value of cos a and we have sin a = 0.25 = 1/4, we'll apply the trigonometric identity
(sin a)^2 + (cos a)^2 = 1 to determine cos a,
We'll recall that a is in 2nd quadrant where the value of cos a is negative.
cos a = - sqrt(1 - sin 2a)
cos a = - sqrt(1 - 1/16)
cos a = - sqrt(15) / 4
We'll substitute cos a by its value in the formula for sin (a/2).
sin (a/2) = sqrt [(1 - sqrt(15)/4)/2]