Find cos(b/2) and cot(b/2) if sin(b) = -sqrt(3)/2 and pi < b < 3pi/2

Question

Find cos(β/2) and cot(β/2) if sin(β) = -√3/2 and π < β < 3π/2

Answer 1

sinB = -sqr3/2, B in quadrant 3

sinB = opposite side over hypotenuse = -sqr3 over 2

arcsin(sqr3/2) = 60 degrees = pi/3 = reference angle

B = pi+ the reference angle = pi+pi/3= 4pi/3= 240 degrees

B/2 = half of 240 = 2pi/3 =240/2 = 120 degrees

tan(B/2) = tan120 = -sqr3

tangent of an angle = opposite side over adjacent side =sqr3 over -1

cot(B/2)= cot120 = 1/tan120 = -1/sqr3 = (-1/3)sqr3

cos(B/2)=cos120 = -cos60 = -1/2

cosine of an angle = adjacent side over hypotenuse

Answer 2