The exact values for the expressions are
sin(θ/2) = 16√65/65
cos(θ/2) = -2√65/65
tan(θ/2) = -8
How determine trigonometric ratios identities of half- angle.
Given that
cscθ = 65/16, 5π/2 < θ < 3π
cscθ = 1/sinθ
sinθ = 16/65
Using Pythagorean identity
cosθ = √1 - sin²θ)
= +-√1 - (16/65)²
= +-√1 - 256/4225
= +-√3969/4225
= +-63/65
θ is in the second quadrant 5π/2 < θ < 3π
θ is negative
θ = -63/65
half-angle identity
sin(θ/2) = +-√(1-coθ)/2
= +-√(1+63/65)/2
= +-√128/65/2
= √256/65
= +-16/√65
= 16√65/65
Sin(θ/2) = 16√65/65 sine is positive in the second quadrant.
cos(θ/2) = +-√(1 + cosθ)/2
= +-√(1 -63/65)/2
= +-√2/65/2
= +-√4/65
= +-2/√65 = 2√65/65
Cosθ is negative in the 2nd quadrant so,
cos(θ/2) = -2√65/65
tan(θ/2) = sinθ/(1 + cosθ)
= (16/65)/(1 - 63/65
= (16/65)/2/65
= 16/65* 65/2
= 8
tanθ is negative in the second quadrant
tan(θ/2) = -8