Find sin x if cos x = 2/3 and x is in quadrant 4

The sine is negative in quadrant 4

adjacent side = 2 and hypotenuse = 3 so opposite side will be - sqrt 5

so sin x = - sqrt5 / 3

Yes you'll use the half angle identities to solve ... it sounds like you understand how to use them just not which sign to use.

This is actually pretty easy. All you have to do is know what quadrant your angle is in and that will determine the sign.

Quadrant1 = all trig functions are positive

Quadrant2 = sine is positive and cosine and tangent are negative

Quadrant3 = tangent is positive and sine and cosine are negative

Quadrant4 = cosine is positive and sine and tangent are negative

since your original angle is in quadrant 4 then the half angle will be in Quadrant 2

so the sine of the half angle is positive and the cosine of the half angle is negative