The exact values of the half angle identities for tanθ = 60/11 and π < θ < 3π/2 is expressed as:
sin(θ/2) = ±√[(1 - cos(180 + tan⁻¹(60/11))/2]
cos(θ/2) = ±√[(1 + cos(180 + tan⁻¹(60/11))/2]
tan(θ/2) = ±√[(1 - cos(180 + tan⁻¹(60/11))/(1 + cos(180 + tan⁻¹(60/11))]
The half angle formula states that:
sin(θ/2) = ±√[(1 - cosθ)/2]
cos(θ/2) = ±√[(1 + cosθ)/2]
tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)]
Given that tanθ = 60/11 and π < θ < 3π/2, then;
θ = tan⁻¹(60/11)
tan is positive at the third quadrant thus
θ = (180 + tan⁻¹(60/11))
θ = 180 + 79.6
θ = 259.6°
Therefore, we can express the exact values of the half angle identities as;
sin(θ/2) = ±√[(1 - cos(180 + tan⁻¹(60/11))/2]
cos(θ/2) = ±√[(1 + cos(180 + tan⁻¹(60/11))/2]
tan(θ/2) = ±√[(1 - cos(180 + tan⁻¹(60/11))/(1 + cos(180 + tan⁻¹(60/11))].