If cos x = 1/4 and cos x < 0, find csc (x/2).

Answer 1

cosx = 1/4 and cosx<0 suggests the problem has a problem

or it got mis-copied

cosx<0 means in the 2nd or 3rd quadrants

so maybe you intended or the problem meant to say cosx =-1/4 with a negative sign

cosx = adjacent side over hypotenuse = -1 over 4

sinx = opposite side over hypotenuse =+ or - sqr15/4= + or - .5sqr15

use the half angle formula, then take the inverse to get csc(x/2)

sinx/2 = + or - square root of [(1-cosx)/2]

= + or - sqr(1/2 - -1/4)

= + or - sqr(3/4) =

= + or - (sqr3)/2

csc(x/2) = + or - 1/sqr(3/4) = + or - 2/sqr3

= + or - 1/(sqr3)/2

= + or - 2/sqr3

BUT maybe the mistake in the problem was that cosx>0, not cosx<0, then x is in quadrant I or IV

with a different solution for csc(x/2)

then cosx=1/4 means sinx is either sqr15/4 or ?

try the same routine, but in other quadrants -