If csc(x) = 9, for 90° < x < 180°, then

Question

sin(x/2) =

cos(x/2) =

tan(x/2) =

Answer 1

csc(x) = hyp/opp

csc(x) = 9/1 therefore the hypotenuse is 9 and the opposite side is 1.

Drawing this in Quadrant II we get:

Solving for the adjacent side "a":

a² + 1² = 9²

a = √(81 - 1) = √80 = 4√5 however, this is in the negative x-direction so must be -4√5

sin(x) = 1/9

cos(x) = -4√5/9

The half angle identities are:

sin(x/2) = ±√[(1 - cos(x))/2]

cos(x/2) = ±√[(1 + cos(x))/2]

tan(x/2) = √[(1 - cos(x))/sin(x)]

We would use the positive results since half of an angle in Q II would be an angle in Q I and all values of sine, cosine, and tangent are positive in Q I.

To get the results, just plug in the values of sin(x) and/or cos(x) into each equation.