©G Dear 2010 – Not to be sold/Free to use Mathematic Extension 1 (Preliminary) Trigonometry Ratios of Half angles Stage 6 - Year 11 Press Ctrl-A ©G Dear 2010 – Not to be sold/Free to use

Half Angles θ 2 2t 1 – t2 If tan = t, then tan θ = tan 2A = 2 tan A Proof Let θ = 2A 2 tan θ 2 1 - tan2 so A = θ 2  tan θ = = 2t 1 – t2 End of Slide

Half Angles θ 2 2t 1 + t2 If tan = t, then sin θ = tan = t = t 1 θ 2 Proof θ 2 1+t2 t sin 2A = 2 sin A cos A 1 Let θ = 2A θ 2  sin θ = 2 sin cos so A = θ 2 1+t2 t 1+t2 1 = 2t 1 + t2 = 2 x x End of Slide

[ ] [ ] Half Angles θ 2 1 – t2 1 + t2 If tan = t, then sin θ = Proof cos 2A = cos2 A – sin2 A t  cos θ = cos2 – sin2 θ 2 1 [ ] 1+t2 1 2 [ ] 1+t2 t 2 Let θ = 2A = - so A = θ 2 = 1 1 + t2 - t2 1 + t2 = 1 – t2 1 + t2 End of Slide