What is the value of sin (a/2) if sin a=0.25? 90<a<180

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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]

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