You can always get all the trig functions of x by knowing one of them and the quadrant:
csc(x) = 6 implies sin(x) = 1/6 If hypotenuse is 6 and vertical is 1, then horizontal is sqrt(35) from Pythagorean theorem and cos(x) = - sqrt(35)/6 (2nd quadrant changes cos to -), tan(x) = -1/sqrt(35)
Now, just use 1/2-angle formulas with the values for the trig functions:
sin(x/2) = sqrt(1/2(1-cos(x)) = sqrt(1/2(1+sqrt(35)/6) = sqrt((6+sqrt(35))/12)
and so on... Note that the half angle will be in the first quadrant, so all trig functions are the positive roots.
Please consider a tutor. Take care.
