Tan θ = 12/5, π < θ < 3π/2

Question

Find cos(θ/2)?

Answer 1

tan θ = 12/5 on the interval (π, 3π/2).

Pythagoras theorem : Hypotenuse² = opposite² + adjacent².

⇒ Hypotenuse = √(12²+ 5²) = √(144 + 25) = √169 = ±13.

Trigonometric identity : cos (θ) = adjacent / hypotenuse.

cos (θ) = ± 5/13.

cos (θ) = - 5/13. (Since, the interval is (π, 3π/2))

Half - angle formula : cos (θ/2) = √[(1 + cos θ)/2].

cos (θ/2) = √[(1 - 5/13)/2]

= √[(13 - 5)/26]

= √[8/26]

= ± 2/√13.

cos (θ/2) = - 2/√13. (Since, the interval is (π, 3π/2))

Therefore, cos (θ/2) = - 2/√13.