Homework Log Tues 5/3 Lesson 8 – 5 Learning Objective: To use product & sum identities to rewrite trig functions Hw: #814 Pg. 516 #6–12 even, 18–24 even
Homework Log Mon 5/2 Lesson 8 – 5 Learning Objective: To use product & sum identities to rewrite trig functions Hw: #813 Pg. 516 #1 – 31 odd
5/2 or 5/3/16 Lesson 8 – 5 Product & Sum Identities Advanced Math/Trig
Learning Objective To use product & sum identities
Product Identities 2 sin 𝛼 cos 𝛽 = sin (𝛼+𝛽) + sin (𝛼−𝛽) 2 cos 𝛼 sin 𝛽 = sin (𝛼+𝛽) − sin (𝛼−𝛽) 2 cos 𝛼 cos 𝛽 = cos (𝛼+𝛽) + cos (𝛼−𝛽) 2 sin 𝛼 sin 𝛽 = cos (𝛼−𝛽) − c𝑜𝑠 (𝛼+𝛽) Given on test – just need to know how to use them
Write as a sum 1. 2sin(3x)cos(2x) 2 sin 𝛼 cos 𝛽 = sin (𝛼+𝛽) + sin (𝛼−𝛽) 2 sin 3𝑥 cos 2𝑥 = sin (3𝑥 +2𝑥)+ sin (3𝑥−2𝑥) = sin(5x) + sin(x)
Write as a sum 2. cos(4y)cos(y) 2 cos 𝛼 cos 𝛽 = cos (𝛼+𝛽) + cos (𝛼−𝛽)
Write as a sum 3. 6 sin 2𝜃 sin 6𝜃 2 sin 𝛼 sin 𝛽 = cos (𝛼−𝛽) − cos (𝛼+𝛽) 6 sin 2𝜃 sin 6𝜃 =3(2 sin 2𝜃 sin 6𝜃 ) = 3 cos (2𝜃 −6𝜃 − cos (2𝜃+6𝜃) ) = 3( cos −4𝜃 − cos (8𝜃) ) cos( −𝜃) = cos 𝜃 = 3 cos 4𝜃 −3 cos 8𝜃
Sum Identities sin 𝑥 + sin 𝑦 =2 sin 𝑥+𝑦 2 cos 𝑥−𝑦 2 sin 𝑥 − sin 𝑦 =2 cos 𝑥+𝑦 2 sin 𝑥−𝑦 2 cos 𝑥 + cos 𝑦 =2 cos 𝑥+𝑦 2 cos 𝑥−𝑦 2 cos 𝑥 − cos 𝑦 =−2 sin 𝑥+𝑦 2 sin 𝑥−𝑦 2 Given on test – just need to know how to use them
Write as a product 4. sin 135 𝑜 + sin 45 𝑜 sin 𝑥 + sin 𝑦 =2 sin 𝑥+𝑦 2 cos 𝑥−𝑦 2 sin 135 𝑜 + sin 45 𝑜 =2 sin 135 𝑜 + 45 𝑜 2 cos 135 𝑜 − 45 𝑜 2 =2 sin 180 𝑜 2 cos 90 𝑜 2 = 2 sin 90 𝑜 cos 45 𝑜
Write as a product 5. cos 3𝜃 − cos 9𝜃 cos 𝑥 − cos 𝑦 =−2 sin 𝑥+𝑦 2 sin 𝑥−𝑦 2 cos 3𝜃− cos 9𝜃 =−2 sin 3𝜃+9𝜃 2 sin 3𝜃−9𝜃 2 =−2 sin 12𝜃 2 sin −6𝜃 2 sin( −𝜃) =− sin 𝜃 = −2 sin 6𝜃 sin (−3𝜃) = 2 sin 6𝜃 sin 3𝜃
Verify 2 sin 4𝜃+2𝜃 2 cos 4𝜃−2𝜃 2 2 cos 4𝜃+2𝜃 2 cos 4𝜃−2𝜃 2 6. sin 4𝜃 + sin 2𝜃 cos 4𝜃 + cos 2𝜃 = tan 3𝜃 2 sin 4𝜃+2𝜃 2 cos 4𝜃−2𝜃 2 2 cos 4𝜃+2𝜃 2 cos 4𝜃−2𝜃 2 = sin 3𝜃 cos 𝜃 cos 3𝜃 cos 𝜃 = sin 3𝜃 cos 3𝜃 = tan 3𝜃
Verify 7. sin 𝐴+𝐵 sin (𝐴−𝐵) = 𝑐𝑜𝑠 2 𝐵− 𝑐𝑜𝑠 2 𝐴 = 1 2 cos 2𝐵− cos 2𝐴 = 1 2 2 𝑐𝑜𝑠 2 𝐵−1 − 2 𝑐𝑜𝑠 2 𝐴−1 = 1 2 2 𝑐𝑜𝑠 2 𝐵−1−2 𝑐𝑜𝑠 2 𝐴+1 = 1 2 2 𝑐𝑜𝑠 2 𝐵−2 𝑐𝑜𝑠 2 𝐴 = 𝑐𝑜𝑠 2 𝐵− 𝑐𝑜𝑠 2 𝐴
Verify 8. sin 2𝑥 − sin 4𝑥 + sin 6𝑥 =4 cos 3𝑥 cos 2𝑥 sin 𝑥 =(2 cos 3𝑥 )( sin 3𝑥 − sin 𝑥 ) Distribute =2 cos 3𝑥 sin 3𝑥 −2 cos 3𝑥 sin 𝑥 = sin 6𝑥−( sin 4𝑥 − sin 2𝑥 ) = sin 6𝑥− sin 4𝑥 + sin 2𝑥
Verify 9. sin 12𝑥 sin 4𝑥 + cos 6𝑥 cos 10𝑥 = cos 6𝑥 cos 2𝑥
Verify 10. cos 7𝑥 sin 𝑥 − sin 2𝑥 cos 6𝑥 =− cos 5𝑥 sin 𝑥 = 1 2 sin 8𝑥 − sin 6𝑥 − 1 2 ( sin 8𝑥 + sin (−4𝑥)) = 1 2 sin 8𝑥 − 1 2 sin 6𝑥 − 1 2 sin 8𝑥 + 1 2 sin 4𝑥 = 1 2 (− sin 6𝑥 + sin 4𝑥 ) = 1 2 ( sin 4𝑥 − sin 6𝑥 ) = 1 2 2 cos 4𝑥+6𝑥 2 sin 4𝑥−6𝑥 2 = cos 5𝑥 sin (−𝑥) =− cos 5𝑥 sin 𝑥
Verify 11. sin 3𝑥 − sin 𝑥 +2 sin 2𝑥 cos 𝑥 =2 sin 3𝑥 =( sin 3𝑥 − sin 𝑥 )+( sin 3𝑥 + sin 𝑥 ) =2 sin 3𝑥 12. cos 9𝑥 − cos 7𝑥 +2 cos 8𝑥 cos 𝑥 =2 cos 9𝑥 =( cos 9𝑥 − cos 7𝑥 )+( cos 9𝑥+ cos 7𝑥 ) =2 cos 9𝑥
Ticket Out the Door Find
Homework #813Pg. 516 #1 – 31 odd