26th November 2013 H U1 O3 Trigonometry Addition Formulae Revision of some Trigonometry from Unit 1Exact Values 2 30 o 30 o 3 2 r π = d 180 1 1 45 o 45 o 2 45 o 45 o 1 1 60 o 60 o 1 2 1 180 x x 180 + x 360 x AutoSave 1
26th November 2013 H U1 O3 Trigonometry Addition Formulae Revision of some Trigonometry from Unit 1Exact Values 2 30 o 30 o 3 2 60 o 60 o 1 2 1 180 x x r π = d 180 1 1 45 o 45 o 2 45 o 45 o 1 1 180 + x 360 x AutoSave 2
26th November 2013 H U1 O3 Trigonometry solve 5sinx = 1 for 0 x 540 o f(x) = 5sinx, g(x) = 1 Algebraic Solution Graphical Solution 180 x x 180 + x 360 x r d π = 180 AutoSave 3
26th November 2013 7sin(2x) = 3.5 0 x 3π H U1 O3 Trigonometry 180 x x r π = d 180 180 + x 360 x AutoSave 4
26th November 2013 H U1 O3 Trigonometry 4 tan (3x+45) o = -2.5 0 o x 180 o 180 x x 180 + x 360 x r π = d 180 AutoSave 5
26th November 2013 H U1 O3 Trigonometry 180 x x Exercise 4H, page 63/64, Q1 d,e,f; Q2 c,d; Q3a,e 180 + x 360 x Exercise 4I, page 65 Q3 AutoSave 6
27th November 2013 H U1 O3 Trigonometry 180 x x Exercise 4H, page 63/64, Q1 d,e,f; Q2 c,d; Q3a,e 180 + x 360 x Exercise 4I, page 65 Q3 AutoSave 7
28th November 2013 H U1 O3 Trigonometry Compound Angles and the Addition Formulae When y=sinα (alpha) is displaced by β (beta) units horizontally its equation becomes y=sin( α ± β) and is called a compound angle. The same is true for y=cos α Compound angle functions can be expanded using the addition formulae: sin(a+b) = sinacosb + cosasinb sin(a-b) = sinacosb - cosasinb cos(a+b) = cosacosb - sinasinb cos(a-b) = cosacosb + sinasinb AutoSave 8
sin(a+b) = sinacosb + cosasinb sin(a-b) = sinacosb - cosasinb cos(a+b) = cosacosb - sinasinb cos(a-b) = cosacosb + sinasinb Expand and simplify cos(x o +60 o ). AutoSave 9
sin(a+b) = sinacosb + cosasinb sin(a-b) = sinacosb - cosasinb cos(a+b) = cosacosb - sinasinb cos(a-b) = cosacosb + sinasinb Show that sin(a+b)=sina.cosb+cosa.sinb for a= πand b= 6 π 3 AutoSave 10
sin(a+b) = sinacosb + cosasinb sin(a-b) = sinacosb - cosasinb cos(a+b) = cosacosb - sinasinb cos(a-b) = cosacosb + sinasinb Find the exact value of sin75 o AutoSave 11
sin(a+b) = sinacosb + cosasinb sin(a-b) = sinacosb - cosasinb Exercise 11B, Q1d, 3a,b, 4d,7 Exercise 11C, Q3, 8a Exercise 11D, Q6a,b, 7, 8 cos(a+b) = cosacosb - sinasinb cos(a-b) = cosacosb + sinasinb AutoSave 12
3rd December 2013 Applications of the Addition Formulae simplifying expressions calculating angles and sides in structures solve equations sin(a±b) = sinacosb ± cosasinb cos(a±b) = cosacosb sinasinb tan x = sin x cos x cos 2 x + sin 2 x = 1 ( and hence: cos 2 x = 1 - sin 2 x and sin 2 x = 1 - cos 2 x) ± SOHCAHTOA Pythagoras' Theorem AutoSave 13
In triangle ABC show that the exact value of sin(a+b) is 2 5 AutoSave 14
On the co-ordinate diagram shown, A is the point (6, 8) and B is the point (12, -5). Angle AOC = p and angle COB = q Find the exact value of sin ( p + q). AutoSave 15
Prove that (sina + cosb) 2 + (cosa - sinb) 2 = 2(1 + sin(a-b)) Exercise 11E, page 193, Q1a, 2a, Exercise 11F, page 194, Q1-4, 7, 9 AutoSave 16
4th December 2013 Applications of the Addition Formulae simplifying expressions calculating angles and sides in structures solve equations sin(a±b) = sinacosb ± cosasinb cos(a±b) = cosacosb sinasinb tan x = sin x cos x cos 2 x + sin 2 x = 1 ( and hence: cos 2 x = 1 - sin 2 x and sin 2 x = 1 - cos 2 x) ± SOHCAHTOA Pythagoras' Theorem Exercise 11E, page 193, Q1a, 2a, Exercise 11F, page 194, Q1-4, 7, 9 AutoSave 17
5th December 2013 Double Angle and Squared Formulae On formula sheet: sin 2α = 2 sinα cosα cos 2α = cos 2 α - sin 2 α = 2 cos 2 α - 1 = 1-2sin 2 α Rearranged from cos2 α : = 2 cos 2 α - 1 cos 2 α = 1 2 (1 + cos 2 α) Rearranged from cos2 α : = 1-2sin 2 α sin 2 α = 1 2 (1 - cos 2 α) (Substitute sin 2 α = 1 - cos 2 α) (Substitute cos 2 α = 1 - sin 2 α) Exercise 11G, page 196, Q1, 4, 5abc, 7, 11abc Exercise 11I, page 201, Q3abcd, 4a AutoSave 18
5th December 2013 Double Angle and Squared Formulae On formula sheet: sin 2α = 2 sinα cosα cos 2α = cos 2 α - sin 2 α = 2 cos 2 α - 1 = 1-2sin 2 α Rearranged from cos2 α : = 2 cos 2 α - 1 cos 2 α = 1 2 (1 + cos 2 α) Rearranged from cos2 α : = 1-2sin 2 α sin 2 α = 1 2 (1 - cos 2 α) (Substitute sin 2 α = 1 - cos 2 α) (Substitute cos 2 α = 1 - sin 2 α) Exercise 11G, page 196, Q1, 4, 5abc, 7, 11abc Exercise 11I, page 201, Q3abcd, 4a AutoSave 19
6th December 2013 Trigonometric Equations (Solving equations involving sin2x and either sinx or cosx) Solve sin2x o = -sinx o for 0 x 360 AutoSave 20
6th December 2013 Trigonometric Equations (Solving equations involving cos2x and cosx) Solve cos2x o = cosx o for 0 x 2π AutoSave 21
6th December 2013 Trigonometric Equations (Solving equations involving cos2x and sinx) Solve cos2x o = sinx o for 0 x 2π AutoSave 22
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