find the exactvalue of sin (theta/2)if cos (theta)=2/3 and 270(deg)<(theta)<360(deg)?

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A few days ago

Anonymous

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If t = tan(theta / 2) then:

cos(theta) = (1 – t^2) / (1 + t^2)

(1 – t^2) / (1 + t^2) = 2 / 3

3(1 – t^2) = 2(1 + t^2)

3 – 3t^2 – 2 – 2t^2 = 0

1 – 5t^2 = 0

t = +/- 1 / sqrt(5)

tan(theta / 2) = +/- 1 / sqrt(5)

As 270 < theta < 360, 135 < theta/2 < 180, making theta/2 a 2nd quadrant angle. That means: tan(theta / 2) = -1 / sqrt(5) sin(theta / 2) = t cos(theta / 2) = t / sec(theta / 2) = t / [ - sqrt(1 + t^2) ] (using the negative root, as sec(theta / 2) is negative in 2nd quadrant) = 1 / ( sqrt(5) * sqrt[1 + 1 / 5] ) = 1 / (sqrt(5 * 6 / 5) = 1 / sqrt(6) = sqrt(6) / 6.

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find the exactvalue of sin (theta/2)if cos (theta)=2/3 and 270(deg)<(theta)<360(deg)?

find the exactvalue of sin (theta/2)if cos (theta)=2/3 and 270(deg)<(theta)<360(deg)?

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