Note 8 Trig Relationships The algebra of trig

+ Note 8: Trig Relationships The algebra of trig is similar to the algebra of real numbers. Example 3 sinθ + 2 sinθ = 5 sinθ ( just like 3 x + 2 x = 5 x ) For more complicated we often use sin 2θ + cos 2θ = 1 earlier lessons) so and sin 2θ = 1 - cos 2θ = 1 – sin 2θ Examples: 1. 3 – 3 sin 2θ = 3( 1 - sin 2θ) = 3 cos 2 θ (because cos 2θ = 1 – sin 2θ) (From

Examples: + 2. sin 2θcosθ+ cos 3θ = cosθ(sin 2θ+ cos 2θ) (Factorising) = cos θ Exercises: 1. Expand Simplify: (sinθ- 2)2 = sin 2θ – 4 sinθ+ 4 (normal expanding of brackets) 2. Factorise: sin 2θ + 5 sinθ+ 6 This is the same method as x 2 + 5 x + 6 = (x + 2)(x + 3) So sin 2θ + 5 sinθ+ 6 = (sinθ + 2)(sinθ+ 3)

+ Page 290 Exercise 13 I

+ Note 9: Double Angle Formulae sin 2 A = 2 sin. Acos. A cos 2 A = cos 2 A – sin 2 A = 2 cos 2 A – 1 = 1 – 2 sin 2 A Example If sin. A = 2/3 find the value of cos 2 A = 1 – 2 sin 2 A = 1 – 2 x (2/3)2 = 1/9

+ Example If θis acute and cos 2θ = ½ find the values of cosθ and sinθ 1. cos 2θ= 2 cos 2 θ – 1 ½ = 2 cos 2 θ – 1 3/4 = cos 2 θ ( -1 and then divide by 2) cosθ = ±√ 3/2 cos θ = √ 3/2 (θis acute, cos is positive) 2. cos 2θ= 1 – 2 sin 2 θ ½ = 1 – 2 sin 2 θ 1/2 = sin 2 θ sinθ = ± 1/2 sinθ = 1/2 (θis acute, sin is positive)

+ OR sin 2θ+ cos 2θ= 1 sin 2θ + (√ 3/2)2 = 1 sin 2θ + 3/4 =1 1/2 = sin 2 θ sinθ = ± 1/2 sinθ = 1/2 (θis acute, sin is positive)

+ Page 293 Exercise 13 J Questions 1 - 5

+ Use appropriate ‘double angle’ formula to simplify: Examples 1. 10 sinθcosθ = 5 (sinθcosθ) = 5 sin 2θ 2. 3 – 6 cos 2 4 A = 3 (1 – 2 cos 2 4 A) = -3(2 cos 2 4 A – 1) = -3 cos 2(4 A) = -3 cos 8 A

+ Page 293 Exercise 13 J Questions 6 - 8

+ Note 10: Tangent Relationships Remember tanθ = sinθ cosθ Example: simplify 1. 4 tanxcosx = 4 sinx x cosx = 4 sinx 2. 3 tanx = 3 sinx x 1 2 sinx cosx 2 sinx = 3 2 cosx 1

+ Page 300 Exercise 13 L. 2 Questions 3 - 5