DERIVING DOUBLEANGLE TRIG IDENTITIES using the sum difference
DERIVING DOUBLE-ANGLE TRIG IDENTITIES using the sum & difference identities that you already know
PART 1 - SINE a. Find by using the sum identity for and simplifying. sinθcosθ +cosθ sinθ =sinθcosθ +sinθ cosθ = 2 sinθcosθ b. You have just discovered the “Double Angle Identity for Sine”. Write the identity in the box below. sin 2θ = 2 sinθcosθ
PART 2 - TANGENT a. Find by using the sum identity for and simplifying. tanθ+tanθ = 2 tanθ 1 - tan 2θ b. You have just discovered the “Double Angle Identity for Tangent”. Write the identity in the box below. 2 tan θ tan 2θ= 1 - tan 2θ
PART 3 - COSINE a. Find by using the sum identity for and simplifying. cosθ – sinθ = cos 2θ – sin 2θ b. You have just discovered the ONE “Double Angle Identity for Cosine”. There are TWO more! cos 2θ= cos 2θ – sin 2θ
PART 3 - COSINE c. To find the second “Double Angle Identity for Cosine”, write the first identity below. Use a Pythagorean substitution to replace sin 2θ in your first identity. Simplify. This is a second “Double Angle Identity for Cosine”. cos 2θ = cos 2θ – sin 2θ = cos 2θ – (1 – cos 2θ) = = cos 2θ – 1 + cos 2θ = 2 cos 2θ – 1 cos 2θ= 2 cos 2θ – 1
PART 3 - COSINE d. To find the third “Double Angle Identity for Cosine”, write the first identity below. Use a Pythagorean substitution to replace cos 2θ in your first identity. Simplify. This is a third “Double Angle Identity for Cosine”. cos 2θ = cos 2θ – sin 2θ = (1 – sin 2θ) – sin 2θ = =1 – sin 2θ = 1 – 2 sin 2θ cos 2θ = 1 – 2 sin 2θ
- Slides: 6