Confused on what should be negative when finding with half angle

Half Angle Formula, Quadrant Rule & Examples

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I am trying to find the work done by gravity for the mass sliding up an inclined plane. I am confused why I don't include the y component of gravity (mgcostheta) in

Is it possible that the value of sin is negative in a half angle

Half Angle Formula, Quadrant Rule & Examples

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Is it possible that the value of sin is negative in a half angle

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Half Angle Formula, Quadrant Rule & Examples

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