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

2. 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

3. 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

4. [ ] [ ] 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