How do you determine whether the sign of a trigonometric function is positive or negative when dealing with half angles?

Consider the quadrant of the original angle and where half that angle would lie.

Half of any positive 1st - or 2nd-quadrant angle will lie in the first quadrant.

Half of any positive 3rd - or 4th-quadrant angle will lie in the second quadrant.

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Half of any negative 1st - or 2nd-quadrant angle will lie in the third quadrant.

Half of any negative 3rd - or 4th-quadrant angle will lie in the fourth quadrant.

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Here, we are using the wording "x-quadrant angle" to mean an angle whose terminal ray lies in the x-quadrant. The measure of that angle may be positive or negative. (Half of - 5π/4 radians will have different trig function values than half of + 3π/4 radians, even though they both have their terminal ray in the 2nd quadrant.)

We are assuming the angle does not exceed 360° (or 2π radians). If it does, you need to adjust the half-angle quadrant accordingly.