cosA = 1/8 with A in QIV, find cscA/2

Answer 1

make your sketch of the Pythagorean triangle in quadrant IV
since cos Ø = adjacent/hypotenuse = x/r
x = 1, r = 8 and x^2 + y^2 = r^2
1 + y^2 = 64
y = ±√63 , but in IV y = -√63
so sinA = -√63/8

we also know cos A = 1 - 2sin^2 (A/2)
2sin^2 (A/2) = 1 - 1/8 = 7/8
sin^2 (A/2) = 7/16
sin (A/2) = ±√7/4 , but if A is in IV, then A/2 is in II and sinA = √7/4

so cscA = 4/√7 or 4√7/7